Borel Measure Nondecreasing and Right Continuous

Elliptic Equations Involving Measures

Laurent Véron , in Handbook of Differential Equations: Stationary Partial Differential Equations, 2004

Theorem 3.43

Let g be a continuous nondecreasing function with finite exponential orders of growth at plus and minus infinity, and λ ∈ $\mathfrak{M}$ ^b (Ω) with decomposition

${\text{λ\hspace{0.17em}=λ}}^{*}+{\text{λ}}_s+{\displaystyle \sum _{j\in J}{c}_j{\delta }_{x_j},}$

λ^*, λ _s being respectively the absolute continuous part and the singular nonatomic part of λ. Let

$\begin{array}{ccc}{J}^{+}=\left\{j\in J:c_j>c_{+}(g)\right\}& and& J^{-}=\left\{j\in J:c_j<c_{-}(g)\right\},\end{array}$

ρ_n be a regularizing kernel and u_n the solution of

(3.165) $\begin{array}{ll}-\Delta u_n+g(u_n)=\text{λ}*{\rho }_n\hfill & in{\mathcal{D}}^{\prime }\text{(}\Omega \text{)},\hfill \\ u_n=0\hfill & on\partial \Omega \text{.}\hfill \end{array}$

Then u_n → u in L ¹(Ω) where u is the solution of

(3.166) $\begin{array}{ll}-\Delta u+g(u)={\text{λ}}^r\hfill & in{\mathcal{D}}^{\prime }\text{(}\Omega \text{)},\hfill \\ u=0\hfill & on\partial \Omega \text{,}\hfill \end{array}$

and

${\text{λ}}^r={\text{λ}}^{*}+{\text{λ}}_s+{\displaystyle \sum _{j\in J\backslash \{J^{+}\cup J^{-}\}}{c}_j{\delta }_{x_j}+}{\displaystyle \sum _{j\in J^{+}}{c}_{+}(g){\delta }_{x_j}+{\displaystyle \sum _{j\in J^{-}}{c}_{-}(g){\delta }_{x_j}}}.$

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Some Elements of the Classical Measure Theory

Endre Pap , in Handbook of Measure Theory, 2002

THEOREM

The following are equivalent:

(a)

There are two bounded non-decreasing functions f ₁, f ₂: ℝ → ℝ such that f = f ₁ − f ₂ on D ∩ Dom f;

(b)

f is of bounded variation on D;

(c)

There ane bounded non-decreasing functions f ₁, f ₂: ℝ → ℝ such that f = f ₁ − f ₂ on D ∩ Domf and ${\text{Var}}_D\left(f\right)=\text{Var}{f}_1+\text{Var}{f}_2.$

Let g: I → ℝ be a function. We assume g is right continuous at every point t ≠ a and that g has finite variation Var _J(g) on every bounded interval J. The variation function of g is the function $\left|g\right|:I\to {ℝ}_{+}$ defined by |g|(t) = Var_[a,t](g), for t ∈ I. Then |g| is increasing and right continuous at every point t ≠ a.

To the function g we associate the finitely additive measure m _g : P[I] → ℝ₊ defined by

and

The measure m_g is positive if and only if g is increasing.

Then to the variation function |g| we associate the positive measure m_|g| , and we have

$\begin{array}{cc}\left|m_g\left(A\right)\right|\le m_{\left|g\right|}\left(A\right),& \text{for}A\end{array}\in \mathcal{P}\left[I\right].$

Since |g| is right continuous at every t ≠ a, the measure m_|g| is σ-additive on P[l] and can be extended uniquely to a σ-additive, positive, finite measure on the ring ℛ(I), still denoted by m_|g| .

From the above inequality, it follows that m_g is σ-additive and with finite variation $\left|m_g\right|\le m_{\left|g\right|}$ on P[I], hence m_g can be extended to a σ-additive measure on the ring ℛ(I), still denoted by m_g , and we still have

$\begin{array}{cc}\left|m_g\left(A\right)\right|\le m_{\left|g\right|}\left(A\right),& \text{for}A\end{array}\in ℛ\left[I\right].$

In fact, we have the equality $\left|m_g\right|\le m_{\left|g\right|}$ .

The positive measure m_|g| can be extended uniquely to a σ-additive measure $\mu :ℬ\left(I\right)\to \left[0,+\infty \right]$ . Moreover, μ is finite on the δ-ring $\mathcal{D}\left(I\right)$ of bounded subsets of I. It follows that the measure m_g can be extended to a σ-additive measure m: $\mathcal{D}\left(I\right)$ → ℝ with finite variation |m| = μ.

If g has bounded variation I, then |g| is bounded on I, the measure m_|g| is bounded on ℛ(I) and its extension μ is bounded on ℬ(I). It follows then that m_g can be extended to a σ-additive, finite measure m : ℬ(I) → ℝ on the Borel σ-algebra, with bounded variation |m| = μ.

We shall continue to denote μ by m_|g| and m by m. Then we still have

$\begin{array}{cc}\left|m_g\right|\left(A\right)\le m_{\left|g\right|}\left(A\right),& \text{for}A\end{array}\in \mathcal{D}\left[I\right].$

or for A ∈ ℬ(I), if g has bounded variation. The σ-additive measure m_g on $\mathcal{D}\left(I\right)$ or ℬ(I) is called the Stieltjes measure on I associated to the function g.

50

The Lebesgue measure. The Lebesgue measure on I is the measure m_g corresponding to the continuous, increasing function g(s) = s, for s ∈ I. In this case we have $m_g]s,t]=m_g[s,t]=m_g[s,t[=m_g]s,t[=t-s$ for every s < t in I.

51

Let −∞ ≤ a < + ∞ and consider an interval I, open to the left, of the form ]a, b] with a < b < + ∞ or of the form ]a. b[. with a < b < + ∞. In particular we can have I =]−∞, + ∞[. This time we denote by R[I] the semiring of the subintervals of I, of the form ]s, t] with a < s. We still denote by ℛ(I) the ring generated by P[I]. The semiring P[I] generates the Borel σ-algebra ℬ(I).

Let g: I → ℝ be a function. We assume g is right continuous at every point t ∈ l and that g has finite variation Var _J (g) on every interval ]a, t] ⊂ I. The variation function of g is the function $\left|g\right|:I\to {ℝ}_{+}$ defined by $\left|g\right|\left(t\right)={\text{Var}}_{\left|a,t\right|}\left(g\right)$ . Then |g| is increasing and right continuous at every point t ≠ a.

To the function g we associate the finitely additive measure m_g: P[I] → ℝ defined by

$m_g]s,t]=g\left(t\right)-g\left(s\right).$

The measure m_g is positive if and only if g is increasing. Then to the variation function |g| we associate the positive measure m_|g| . As in the previous case, the measure m_g is σ-additive, m_g is σ-additive, with finite variation |m_g| = m_g . We can extend m_|g| to a σ-additive μ:ℝ(I) → [0,+ ∞] and then we can extend m_g to a σ-additive measure m: $\mathcal{D}\left(I\right)$ → ℝ with finite variation |m| = μ. If g has bounded variation on I, then m_g can be extended to a σ-additive measure m: ℬ(I) → ℝ on the Borel σ-algebra, with bounded variation |m| = μ. As in the previous case, we continue to denote μ by m_|g| and m by m_g . The σ-additive measure m_g on $\mathcal{D}\left(I\right)$ or ℬ(I) is called the Stieltjes measure on I associated to the function g.

52

The Lebesgue measure on I is the positive measure associated to the increasing, continuous function g(s) = s, for s ∈ I. In this case, we still have

$m_g]s,t]=m_g[s,t]=m_g[s,t]=m_g]s.t[=t-s.$

53

Remark. If I is an interval closed to the left or open to the left and if g: I → ℝ is a function with finite variation, we can consider the right continuous function g₊ : I → ℝ defined by g₊(a) = g(a), if a ∈ I and g₊(t) = g(t+), if a <t, and proceed as above.

54

As a consequence of the Monotone Class Theorem we have the following result: if μ, υ are two measures on ℬ(ℝ^r), where r ≥ 1, both defined, and agreeing, on all intervals of the form

$]-\infty ,a]=\left\{x:x\le a\right\}=\left\{\left({\xi }_1,\mathrm{\dots }{\xi }_r\right):{\xi }_i\le {\alpha }_i\text{\hspace{0.17em}for\hspace{0.17em}every}i\le r\right\},$

for $a=\left({\alpha }_1,\dots ,{\alpha }_r\right)\in {ℝ}^r$ , and $\mu \left({ℝ}^r\right)<\infty$ then μ and υ are equal on all the Borel subsets of ℝ _r .

55

THEOREM (Image measures). Let (S, Σ, μ) be a measure space, Y any set, and ${\phi }^{-1}:S\to Y$ a function. Set

$\begin{array}{cc}\mathcal{T}=\left\{F:F\subseteq Y,{\phi }^{-1}\left(F\right)\in \Sigma \right\},& v\left(F\right)=\mu \left({\phi }^{-1}\left(F\right)\right)\end{array},foreveryF\in \mathcal{T}.$

Then (Y, $\mathcal{T}$ , υ) is a measure space, υ is called the image measure $\mu {\phi }^{-1}$ .

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Models for Spectral Analysis—The Univariate Case

Lambert H. Koopmans , in The Spectral Analysis of Time Series, 1995

The Spectral Representation of the Autocovariance Function

Wiener established the existence of a bounded nondecreasing function F(λ), called the spectral distribution function, such that

(2.4) $C\left(\tau \right)={\int }_{-\infty }^{\infty }{e}^{i\lambda \tau }F\left(d\lambda \right)\cdot$

This important expression is called the spectral representation of the autocovariance function. The spectral distribution function determines a measure F(A) called the spectral distribution of the time series and (2.4) is an integral with respect to this measure. This theory can be outlined exclusively in terms of the distribution function without explicitly introducing the idea of measure. However, to do so has the disadvantage that simple and intuitive properties have rather clumsy and unpleasant notational expressions. Simply think of the spectral distribution F(A) as the amount of power in the harmonic components of the time series with frequencies in the set A. The spectral distribution and spectral distribution function are related by the expression

$F\left(\lambda \right)=F\left(\left(-\infty ,\lambda \right]\right)\cdot$

Thus, for example, F(λ) is the power for frequencies less than or equal to λ. The symbol (a, b] denotes the interval {x: a < x ≤ b}. This notation for intervals will be used throughout the book with the exclusion and inclusion of endpoints being indicated by round and square brackets respectively.

The integral (2.4) can be reduced to more familiar terms by the Lebesgue decomposition of F(A) [see Grenander and Rosenblatt (1957, p. 35)]. For models of practical interest this measure can be expressed as the sum of two components

(2.5) $F\left(A\right)=F_{\text{d}}\left(A\right)+F_{\text{c}}\left(A\right)\cdot$

The discrete spectral distribution F _d(A) is completely characterized by a function p(λ), called the spectral function, which has the property that p(λ) ≥ 0 for all λ and p(λ) > 0 only among a "discrete" set of frequencies …, λ₋₁, λ₀, λ₁, …, where λ_−j = −λ _j (thus λ₀ = 0). Then,

(2.6) $F_{\text{d}}\left(A\right)=\underset{{\lambda }_j\in A}{\Sigma }p\left({\lambda }_j\right)\cdot$

The value of p(λ) at frequency λ is the spectral mass concentrated at that frequency and, thus, is related to the spectral distribution by the expression

$p\left(\lambda \right)=F\left(\left\{\lambda \right\}\right),$

where {x} denotes the set consisting of the single element x.

The continuous component F_c (A) is determined by the derivative of the spectral distribution function f(λ) = F′(λ), where f(λ) is called the spectral density function, and the amount of continuous power or continuous spectral mass in a set of frequencies A is given by

(2.7) $F_{\text{c}}\left(A\right)={\int }_{A}f\left(u\right)du\cdot$

Since the derivative of a nondecreasing function is nonnegative, the spectral density function has the property

$\begin{array}{ccc}f\left(\lambda \right)\ge 0& \text{for}& \text{all}\lambda \cdot \end{array}$

It is sometimes useful to think of the continuous power in a set A as being given by the area under the curve y = f(λ) over the set A.

The "spectral mass" or power in a set of frequencies A when both discrete and continuous components are present in the spectrum is

(2.8) $F\left(A\right)=\underset{{\lambda }_j\in A}{\Sigma }p\left({\lambda }_j\right)+{\int }_{A}f\left(\lambda \right)d\lambda \cdot$

The spectral representation of the autocovariance (2.4) can then be represented in the form

$C\left(\tau \right)=\underset{k=-\infty }{\overset{\infty }{\Sigma }}{e}^{i{\lambda }_k\tau }p\left({\lambda }_k\right)+{\int }_{-\infty }^{\infty }{e}^{i\lambda \tau }f\left(\lambda \right)d\lambda \cdot$

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Theory of Hp Spaces

In Pure and Applied Mathematics, 1970

Corollary.

Let φ(x ) ≥ 0 be a convex nondecreasing function of x ≥ 0, such that φ(x) = o(x log x)as x → ∞. Then there exists a harmonic function n(z), with conjugate v(z), such that

${\displaystyle {\int }_0^{2\pi }\phi \left(\left|u\left(r{e}^{i\theta }\right)\right|\right)}d\theta$

is bounded for r < 1, while v ∉ h ¹.

PROOF. Choose an integrable function U(t) ≥ 0 on [0, 2π] such that φ(U(t)) is integrable but U(t) log⁺ U(t) is not. (We will show in a moment that this is possible.) Let u(z) be the Poisson integral of U(t). Then u(z) > 0, and an application of Jensen's inequality (Chapter 2, Exercise 4) shows that ∫ φ(u(re ^iθ )) dθ is bounded. By the preceding theorem, v ∈ h ¹ would imply u ∈ h log⁺ h, which is not the case (by Fatou's lemma). Thus v ∉ h ¹.

It remains to show that a function U(t) can be constructed with the required properties. Suppose, more generally, that φ(x) and ψ(x) are any continuous, nonnegative, nondecreasing functions (convex or not), defined on 0 ≤ x < ∞, such that ψ(x) → ∞ and φ(χ)/ψ(x) → 0 as x → ∞. Choose increasing numbers x_n (n = 1, 2, …) such that ψ(x_n ) = 2 ⁿ . Then 2^–n φ(x_n ) → 0; select a subsequence {n_k } for which

${\displaystyle \sum _{k=1}^{\infty }{2}^{-n_k}\phi \left(x_{n_{\kappa }}\right)}<\infty .$

Now define t ₀ = 0 and $t_k=t_{k-1}+2^{-nk},k\ge 1.$ k ≥ 1. Notice that 0 < t ₁ < t ₂ < ⋯ ≤ 1. Let T = lim t_k . Finally, let U(t) be the step function with the value $x_{nk}$ in the interval t _k–1 < t < t_k and with U(t) = 0 in T ≤ t < 2π. Observe that

$\begin{array}{l}{\displaystyle {\int }_0^{2\pi }\phi \left(U\left(t\right)\right)}dt={\displaystyle \sum _{k=1}^{\infty }\left(t_k-t_{k-1}\right)}\phi \left(x_{n_k}\right)+\left(2\pi -T\right)\phi \left(0\right)\\ {\displaystyle \sum _{k=1}^{\infty }{2}^{-n_k}}\phi \left(x_{n_k}\right)+\left(2\pi -T\right)\phi \left(0\right)<\infty .\end{array}$

On the other hand, ψ(U(t)) is not integrable because $2^{-nk}\psi \left(x_{nk}\right)=1.$

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A Geometric Interpretation of Pontryagin's Maximum Principle1

EMILIO ROXIN , in International Symposium on Nonlinear Differential Equations and Nonlinear Mechanics, 1963

Proof

As δ(x ₀ , ∈) is a nondecreasing function of ∈ for ∈ > 0, there exists the limit

$l=\underset{\in \to \infty }{\lim }\delta (x_0,\in ).$

Let us take the sequences of positive numbers ∈ _i → 0, ρ _i → 0 for i = 1, 2, 3,…. For any i we can select a value u_i ∈ U and a value x_i such that ‖ x ₀ – x_i ‖ ≦ ∈ _i and

$\delta (x_0,{\in }_i)\geqq \Vert f(x_i,u_i)-f(x_0,u_i)\Vert \geqq \delta (x_0,{\in }_i)-{\rho }_i.$

As i → ∞, ρ _i → 0 and therefore

$\Vert f(x_i,u_i)-f(x_0,u_i)\Vert \to l.$

According to (6.4′), x_i → x ₀. Since U is compact, we can select a subsequence such that the lim u_i = u ₀ exists; we shall suppose that i = 1,2, 3,…, refers to that subsequence. Hence, the continuity of f(x, u) for x = x ₀ assumes that

$l=\underset{i\to \infty }{\lim }\Vert f(x_i,u_i)-f(x_0,u_i)\Vert =0.$

With these elements we can prove for rather general conditions, what in the simplest cases is quite obvious, namely that "in first approximation" the reachable points from a given x ₀, are those which are inside the cone F[x ₀].

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Systems of Riemann–Liouville fractional differential equations with uncoupled integral boundary conditions

Johnny Henderson , Rodica Luca , in Boundary Value Problems for Systems of Differential, Difference and Fractional Equations, 2016

4.3.1 Presentation of the problem

We consider the system of nonlinear ordinary fractional differential equations

(S⁰) $\begin{array}{l}\hfill \left\{\begin{array}{l}{D}_{0+}^{\alpha }u(t)+a(t)f(v(t))=0,t\in (0,1),\\ D_{0+}^{\beta }v(t)+b(t)g(u(t))=0,t\in (0,1),\end{array}\right.\end{array}$

with the uncoupled integral boundary conditions

(BC⁰) $\begin{array}{l}\hfill \left\{\begin{array}{l}u(0)=u^{\prime }(0)=\cdots =u^{(n-2)}(0)=0,u(1)={\int }_0^{1}u(s)\mathrm{d}H(s)+a_0,\\ v(0)=v^{\prime }(0)=\cdots =v^{(m-2)}(0)=0,v(1)={\int }_0^{1}v(s)\mathrm{d}K(s)+b_0,\end{array}\right.\end{array}$

where n − 1 < α ≤ n, m − 1 < β ≤ m, $n,m\in \mathbb{N}$ , n, m ≥ 3, $D_{0+}^{\alpha }$ and $D_{0+}^{\beta }$ denote the Riemann–Liouville derivatives of orders α and β, respectively, the integrals from (BC⁰) are Riemann–Stieltjes integrals, and a ₀ and b ₀ are positive constants.

By using the Schauder fixed point theorem (Theorem 1.6.1), we shall prove the existence of positive solutions of problem (S⁰)–(BC⁰). By a positive solution of (S⁰)–(BC⁰), we mean a pair of functions $(u,v)\in C([0,1];{\mathbb{R}}_{+})\times C([0,1];{\mathbb{R}}_{+})$ satisfying (S⁰) and (BC⁰) with u(t) > 0,v(t) > 0 for all t ∈ (0,1]. We shall also give sufficient conditions for the nonexistence of positive solutions for this problem.

We present the assumptions that we shall use in the sequel:

(J1)

$H,K:[0,1]\to \mathbb{R}$ are nondecreasing functions, ${\mathrm{\Delta }}_1=1-{\int }_0^1{s}^{\alpha -1}\mathrm{d}H(s)>0$ , and ${\mathrm{\Delta }}_2=1-{\int }_0^1{s}^{\beta -1}\mathrm{d}K(s)>0$ .

(J2)

The functions $a,b:[0,1]\to [0,\infty )$ are continuous, and there exist t ₁, t ₂ ∈ (0,1) such that a(t ₁) > 0, b(t ₂) > 0.

(J3)

$f,g:[0,\infty )\to [0,\infty )$ are continuous functions, and there exists c ₀ > 0 such that $f(u)<\frac{c_0}{L}$ , $g(u)<\frac{c_0}{L}$ for all u ∈ [0,c ₀], where $L=max\{{\int }_0^{1}a(s)J_1(s)\mathrm{d}s,{\int }_0^{1}b(s)J_2(s)\mathrm{d}s\},$ and J ₁ and J ₂ are defined in Section 4.1.2.

(J4)

$f,g:[0,\infty )\to [0,\infty )$ are continuous functions and satisfy the conditions ${lim}_{u\to \infty }\frac{f(u)}{u}=\infty ,{lim}_{u\to \infty }\frac{g(u)}{u}=\infty .$

Under assumption (J1), we have all auxiliary results in Lemmas 4.1.4–4.1.9 from Section 4.1.2. Besides, by (J2) we deduce that ${\int }_0^{1}a(s)J_1(s)\mathrm{d}s>0$ and ${\int }_0^{1}b(s)J_2(s)\mathrm{d}s>0$ —that is, the constant L from (J3) is positive.

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Handbook of Differential Equations

J. Dávila , in Handbook of Differential Equations: Stationary Partial Differential Equations, 2008

Theorem 1.3(Brezis et al. [19]).

Assume g : [0, ∞] → ∞ is a C ¹ convex nondecreasing function. If (1.1) has a weak solution w then for any initial condition u ₀∈ L ^∞(Ω), 0 ≤ u ₀≤ w the solution to (1.69) is global in time.

Peral and Vazquez [100] considered also the parabolic problem (1.69) with the exponential nonlinearity in Ω = B ₁and with λ = 2(N –2), since for this parameter U (x) =−2 log | x | is a weak solution of the stationary problem. They are interested in singular initial conditions and hence they work with the following notion of weaksolution: $u\in C\left(\left(0,\infty \right);W_0^{1,2}\left(B_1\right)\right)$ such that $u_t,\Delta u,e^u\in L^1\left(\left[\tau ,T\right]\times B_1\right)$ for all 0 <τ < T <+∞,equation (1.69) holds a.e. and u (t, ·) → u ₀in L ²(B ₁) as t → 0. First they take an initial condition u ₀satisfying 0 ≤ u ₀(x) ≤ U (x). They show that (1.69) possesses a minimal and a maximal solution u satisfying 0 ≤ u (t, x) ≤ U (x). Moreover it becomes classical for t > 0. They show that if 3 ≤ N ≤ 9 then any solution satisfying the previous conditions converges to the minimal solution u _λas t → +∞. If N ≥ 10 then u (t, ·) → U as t → +∞. These authors also study the possibility of having solutions of the parabolic problem above the singular solution U and establish the following

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Stochastic Processes

In Stochastic Processes, 2004

1.2.4. Probability Distribution and Probability Density

This section introduces in a simple way the skeletal structure of the basic mathematical tools related to probability distribution and probability density functions. They play an important role in many areas (control, optimization, image signal processing, reliability, chemical engineering, mineral industry, etc.).

Probability Distribution

The probability distribution gives the relationship (correspondence) between all possible values (realizations) of a given random variable and their associated probabilities. Certainly, a table is the most simple form for the representation of this correspondence [65]. Let x _i (i = 1, …, K) represent the possible values of the random variable X, and p _i (i = 1, …, K) the corresponding probabilities. We can derive the following table:

$\frac{x_i^{}}{P_i^{}}|\frac{x_1{x}_2\mathrm{...}{x}_k}{P_1{P}_2\mathrm{...}{P}_k}$

When the value of K is relatively large, this representation is not useful in practice. It is better from an engineering point of view to have a graphical representation as shown in Figure 1.13. In this figure the values x _i (i = 1, …, K) are reported on the abscissae, and their associated probabilities p _i (i = 1, …, K) are reported on the ordinates. The extremities of the vertical lines representing the probabilities have been connected by segments. Hence, we obtain a polygonal representation.

Figure 1.13. Graphical representation

Observe that K can be large, or tend to infinity; and a continuous random variable has an infinity of values in a given interval. We shall now be concerned with the (cumulative) probability distribution of a continuous variable X which will be denoted by F(x), and defined as follows:

$F\left(x\right)=Pr\left(X<x\right).$

The distribution function contains all the information which is relevant to probability theory.

Example 6 Let the probability distribution of a random variable X be

$F_X\text{(}x\text{)=}\{\begin{array}{l}\text{0}forx\text{<0}\hfill \\ xfor\text{0}\le x\text{<1}\hfill \\ \text{1}forx\ge \text{1}\text{.}\hfill \end{array}$

Find the probability distribution of the random variable Z = √X.

Solution We have to consider only the case when x ∈ [0, 1]. It follows that:

$F_Z\text{(}x\text{)=Pr(}Z\le x\text{)=Pr(}\sqrt{X}\le x\text{)=Pr(}X\le x^{\text{2}}\text{)=}{F}_X\text{(}{x}^{\text{2}}\text{)}.$

The probability distribution has the following properties:

1.

F(x ) is a nondecreasing function.

2.

lim _x→−∞ F(x) = 0, and lim _x→+∞ F(x) = 1.

3.

F(x) is continuous at least from the left.

These properties are evident. In fact, if x increases, Pr(X<x) increases also and vice versa, and the probability of the event X +∞ (X < − ∞) is equal to 1 (0).

Example 7 (Lemma 4.1.1 in [ 27 ]) Let the probability distribution F(t) of the random variable Y be continuous. Show that the random variable Z = F(Y) is uniformly distributed over the interval [0, 1].

Solution In view of the definition of the probability distribution F(t) = Pr(Y<t), we have,

$\text{Pr(}Z\text{<}x\text{)=Pr(}Y\text{<}{F}^{-1}\text{(}x\text{))=}F\text{(}{F}^{-1}\text{(}x\text{))=}x,$

where F ⁻¹(x) represents the inverse of F(t), for 0 ≤ x ≤ 1.

Probability Density

Now, we are interested in the probability that a continuous random variable X belongs to the interval [x, x + Δx]. It is clear that this probability is equal to the variation of the probability distribution in this interval, i.e.,

$\text{Pr(}x\text{<}X\le x\text{+Δ}x\text{)=}F\text{(}x\text{+Δ}x\text{)−}F\text{(x)}.$

Let us assume that F(x) is continuous and is differentiable, and consider the mean of this probability towards the length unit. If Δx → 0, we obtain

(1.7) $\underset{\text{Δ}x\to \text{0}}{\lim }\frac{F\text{(}x\text{+Δ}x\text{)−}F\text{(}x\text{)}}{\text{Δ}x}\text{=}\frac{dF\text{(}x\text{)}}{dx}\text{=}f\text{(}x\text{)}.$

where f(x) represents the probability density function (pdf). In summary, every real valued function which is nonnegative (Pr(·) ≥ 0), integrable over the whole real axis, and satisfies (1.7) is the probability density of a random variable X.

The pdf is useful for calculating the probability that a random variable X belongs to a given interval, say (a, b). This probability is given by

$\text{Pr(}a\le X\le b\text{)=}\underset{b}{\overset{a}{{{\displaystyle \int }}^{}}}f\text{(}x\text{)}dx.$

In other words, this probability is equal to the area between the curve f(x) and the x-axis.

In many situations, the probability density function is a priori unknown. One possible way to obtain to a mathematical expression of f(x) is to consider a model of the form

$f\text{(}x\text{)=}{\displaystyle \sum _{i=1}^{\infty }{\alpha }_i{\varphi }_i\text{(}x\text{)}},$

where ϕ _i (x) (i = 1, 2, …) are a priori known functions. A truncated series of m terms

$f\text{(}x\text{)=}{\displaystyle \sum _{i=1}^m{\alpha }_i{\varphi }_i\text{(x)}}$

can then be used as an approximation of the true function.

Remark 3 (Modeling) Statistical analysis can be used to fit unknown model parameters, and to evaluate the uncertainty related to the fitted model, as well as to compare several candidate functions ϕ _i (x). One common approach is to use the optimization theory to carry out least-squares estimates for the parameters α_i .

Two Random Variables

Consider a system consisting of two random variables X and Y. As for single random variables, we shall define the characteristics of the system (X, Y).

•

Probability distribution:

$F\text{(}x\text{,}y\text{)=Pr((}X\le x\text{)(}Y\le y\text{))}$

•

Probability density: expression (1.7) is extended as follows:

(1.8) $\begin{array}{ll}\text{Pr((}X\text{,}Y\text{)}\in \text{Λ)}\hfill & \text{=}F\text{(}x\text{+Δ}x\text{,}y\text{+Δ}y\text{)−}F\text{(}x\text{+Δ}x\text{,}y\text{)}\hfill \\ \hfill & \text{−}F\text{(}x\text{,}y\text{+Δ}y\text{)−}F\text{(}x\text{,y)}\hfill \end{array}$

where Λ represents a rectangle of dimension Δx and Δy. Expression (1.8) leads to

$\begin{array}{cc}\underset{\text{Δ}x\to \text{0}\text{.Δ}y\to \text{0}}{\lim }\frac{\text{Pr((}X\text{,}Y\text{)}\in \text{Λ)}}{\text{Δ}x\text{Δ}y}& \text{=}\underset{\text{Δ}x\to \text{0}\text{.Δ}y\to \text{0}}{\lim }\frac{\text{1}}{\text{Δ}x\text{Δ}y}\{F\text{(}x\text{+Δ}x,y\text{+Δ}y\text{)}\text{.}\\ & \text{−}F\text{(}x\text{+Δ}x\text{,}y\text{)−}F\text{(}x\text{,}y\text{+Δ}y\text{)−}F\text{(}x\text{,}y\text{)}}\text{.}\end{array}$

If the function F(x, y) is continuous and differentiable, we get

$f\text{(}x,y\text{)=}\frac{{\partial }^{\text{2}}F\text{(}x\text{,}y\text{)}}{\partial x\partial y}.$

.

•

Moment of order s, k:

$m_{s\text{,}k}\text{=}E\text{{}{X}^s{Y}^k\text{},}$

where E{·} denotes the expectation operator (see next section).

•

Covariance:

$K_{xy}\text{=}Cov\text{(}X\text{,}Y\text{)=}E\text{{(}X\text{−}{m}_x\text{)(}Y\text{−}{m}_y\text{)},}$

where m _x and m _y are the mean of X and Y, respectively. The covariance characterizes the dispersion of random variables and the degree of their dependency. Notice that for independent random variables the covariance is equal to zero ⁴ . Observe that if the dispersion of a given random variable, say X, is small(the random variable is close to its mean), the covariance will be small even if the random variables X and Y are closely dependent. In order to characterize only the dependency between random variables, the coefficient of correlation is introduced.

•

Coefficient of correlation:

$r_{xy}\text{=}\frac{K_{xy}}{{\sigma }_x{\sigma }_y},$

where σ _x ² and _y ² represent the variance of X and Y, respectively.

In the subsequent paragraphs, two operators of profound importance are discussed, namely the expectation and the conditional mathematical expectation.

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Topics of Functional Analysis

Alexander S. Poznyak , in Advanced Mathematical Tools for Automatic Control Engineers: Deterministic Techniques, Volume 1, 2008

Proof

For x > y from (17.80) it follows that f (x) is a nondecreasing function and, in view of (17.81), there exist numbers a and b such that a > b, f (a) > 0 and f (b) > 0. Then, considering f (x) on [a, b], by the theorem on intermediate values, there exists a point $f\left(\xi \right)=0$ such that f (ξ) = 0. If (18.72) is fulfilled, then f (x) is a monotonically increasing function and the root of the function f (x) is unique.

The following definitions and theorems represent the generalization of this lemma to functional spaces and nonlinear operators.

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Integral and Finite Difference Inequalities and Applications

B.G. Pachpatte , in North-Holland Mathematics Studies, 2006

Theorem 2.3.5

Let u(x,y) f(x,y)∈C $\left(R_{+}^2,R_{+}\right)$ , h(x,y,s,t) ∈ C $\left(R_{+}^4,R_{+}\right)$ for 0 ≤ s ≤ x > ∞, 0≤ t ≤ y > ∞ and c ≥ 0, p > 1 are real constants.

(k ₁) Let g ∈ C (R₊,R₊ ) be a nondecreasing function, g(u) > 0 for u > 0. If

(2.3.54) $\begin{array}{l}{u}^p\left(x,y\right)\le c+{\displaystyle \underset{0}{\overset{x}{\int }}{\displaystyle \underset{0}{\overset{y}{\int }}[f\left(s,t\right)g\left(u\left(s,t\right)\right)}}\\ +{\displaystyle \underset{0}{\overset{s}{\int }}{\displaystyle \underset{0}{\overset{t}{\int }}h\left(s,t,\sigma ,\eta \right)g\left(u\left(\sigma ,\eta \right)\right)d\eta d\sigma }}]dtds,\end{array}$

for x,y ∈ R₊ , then for 0 ≤ x ≤ x₁,0 ≤ y ≤ y₁;x,x₁,y,y₁ ∈ R₊ ,

(2.3.55) $u\left(x,y\right)\le {\left\{G^{-1}\left[G\left(c\right)+A\left(x,y\right)\right]\right\}}^{\frac{1}{p}},$

where

(2.3.56) $A(x,y)={\displaystyle \underset{0}{\overset{x}{\int }}{\displaystyle \underset{0}{\overset{y}{\int }}[f(s,t)+{\displaystyle \underset{0}{\overset{s}{\int }}{\displaystyle \underset{0}{\overset{t}{\int }}h(s,t,\sigma ,\eta )d\eta d\sigma }}]dtds,}}$

(2.3.57) $G\left(r\right)={\displaystyle \underset{r_0}{\overset{r}{\int }}\frac{ds}{g\left(s^{\frac{1}{p}}\right)},r>0,}$

r₀ > 0 is arbitrary, G⁻¹ is the inverse function of G and x ₁, y₁ ∈ R₊ are chosen so that

$G\left(c\right)+A\left(x,y\right)\in Dom\left(G^{-1}\right),$

for all x, y lying in the intervals 0 ≤ x ≤ x₁ , 0 ≤ y ≤ y₁ of R₊.

(k₂ ) If

(2.3.58) $\begin{array}{l}{u}^p\left(x,y\right)\le c+{\displaystyle \underset{0}{\overset{x}{\int }}{\displaystyle \underset{0}{\overset{y}{\int }}[f\left(s,t\right)u\left(s,t\right)}}\\ +{\displaystyle \underset{0}{\overset{s}{\int }}{\displaystyle \underset{0}{\overset{t}{\int }}h\left(s,t,\sigma ,\eta \right)u\left(\sigma ,\eta \right)d\eta d\sigma }}]dtds,\end{array}$

for x,y ∈ R₊, then

(2.3.59) $u\left(x,y\right)\le {\left\{c^{\frac{p-1}{p}}+\left(\frac{p-1}{p}\right)A\left(x,y\right)\right\}}^{\frac{1}{p-1}},$

for x,y ∈ R₊ , where A(x,y) is defined by (2.3.56).

Proof. (k₁) Let c > 0 and define a function z(x,y) by the right hand side of (2.3.54). Then z(0,y) = z(x,0i = c, u(x,y) ≤ ${\left(z\left(x,y\right)\right)}^{\frac{1}{p}}$ and

(2.3.60) $\begin{array}{l}{D}_{1}z\left(x,y\right)={\displaystyle \underset{0}{\overset{y}{\int }}\left[f\left(x,t\right)g\left(u\left(x,t\right)\right)+{\displaystyle \underset{0}{\overset{x}{\int }}{\displaystyle \underset{0}{\overset{t}{\int }}h\left(x,t,\sigma ,\eta \right)g\left(u\left(\sigma ,\eta \right)\right)d\eta d\sigma }}\right]}dt\\ \le {\displaystyle \underset{0}{\overset{y}{\int }}\left[f\left(x,t\right)g\left(z{\left(x,t\right))}^{\frac{1}{p}}\right)+{\displaystyle \underset{0}{\overset{x}{\int }}{\displaystyle \underset{0}{\overset{t}{\int }}h\left(x,t,\sigma ,\eta \right)g{\left(u\left(\sigma ,\eta \right)\right)}^{\frac{1}{p}}d\eta d\sigma }}\right]}dt\\ \le g\left({\left(z\left(x,y\right)\right)}^{\frac{1}{p}}\right){\displaystyle \underset{0}{\overset{y}{\int }}\left[f\left(x,t\right)+{\displaystyle \underset{0}{\overset{x}{\int }}{\displaystyle \underset{0}{\overset{t}{\int }}h\left(x,t,\sigma ,\eta \right)d\eta d\sigma }}\right]dt.}\end{array}$

From (2.3.57) and (2.3.60) we observe that

(2.3.61) $\begin{array}{l}{D}_{1}G\left(z\left(x,y\right)\right)=\frac{D_{1}z\left(x,y\right)}{g{\left(\left(z9x,y\right)\right)}^{\frac{1}{p}}}\\ \le {\displaystyle \underset{0}{\overset{y}{\int }}\left[f\left(x,t\right)+{\displaystyle \underset{0}{\overset{x}{\int }}{\displaystyle \underset{0}{\overset{t}{\int }}h\left(x,t,\sigma ,\eta \right)d\eta d\sigma }}\right]}dt.\end{array}$

Keeping y fixed in (2.3.61), setting x = s and integrating with respect to s from 0 to x and using the fact that z(0, y) = c, we have

(2.3.62) $G\left(z\left(x,y\right)\right)\le G\left(c\right)+A\left(x,y\right).$

Now substituting the bound on z(x,y) from (2.3.62) in $u\left(x,y\right)\le {\left(z\left(x,y\right)\right)}^{\frac{1}{p}}$ we obtain the desired bound in (2.3.55). The proof of the case when c ≥ 0 can be completed as mentioned in the proof of Theorem 1.3.3. The subdomain 0 ≤ x v x₁ , 0 ≤ y ≤ y ₁ is obvious.

(k₂ ) The proof is similar to that of given in Theorem 1.3.3 and we omit it here.

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