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Changing a Function on a Null Set

#4915

Analysis

Statement

Let $E \subset \mathbb{R}^n$ be Lebesgue measurable. Let $f: E \to [0,\infty]$ and $g: E \to [0,\infty]$ be Lebesgue [measurable functions](/page/Measurable%20Functions) such that $f=g$ for $\mathcal{L}^n$-a.e. $x \in E$. Then \begin{align*} \int_E f\, d\mathcal{L}^n = \int_E g\, d\mathcal{L}^n. \end{align*}

Description

This theorem states that changing a measurable function only on a Lebesgue null set preserves measurability, capturing the almost-everywhere nature of measurable functions.

Analysis

Linearity and Positivity

#4904

Probability & Statistics

Statement

Let $X,Y\in L^1(\Omega,\mathcal F,\mathbb P)$, let $a,b\in\mathbb R$, and let $\mathcal G\subset\mathcal F$ be a sub-$\sigma$-algebra. Then \begin{align*} \mathbb E[aX+bY\mid\mathcal G]=a\mathbb E[X\mid\mathcal G]+b\mathbb E[Y\mid\mathcal G] \end{align*} $\mathbb P$-a.e. If $X\ge0$ $\mathbb P$-a.e., then \begin{align*} \mathbb E[X\mid\mathcal G]\ge0 \end{align*} $\mathbb P$-a.e.

Description

Conditional expectation is linear in integrable random variables and preserves almost sure nonnegativity under conditioning.

Probability & Statistics

Doob-Dynkin Representation

#4902

Probability & Statistics

Statement

Let $Z:(\Omega,\mathcal F)\to(E,\mathcal E)$ be a random variable taking values in a standard Borel space, and let $Y:(\Omega,\mathcal F)\to(\mathbb R,\mathcal B(\mathbb R))$ be a $\sigma(Z)$-measurable random variable. Then there exists a measurable function $g:E\to\mathbb R$ such that \begin{align*} Y=g(Z) \end{align*} $\mathbb P$-a.e.

Description

A sigma(Z)-measurable real random variable can be represented almost surely as a measurable function of the random variable Z.

Probability & Statistics

Theorem #4905

#4905

Probability & Statistics

Theorem #4907

#4907

Probability & Statistics

$L^p$ Contraction

#4906

Probability & Statistics

Statement

Let $1\le p<\infty$, let $X\in L^p(\Omega,\mathcal F,\mathbb P)$, and let $\mathcal G\subset\mathcal F$ be a sub-$\sigma$-algebra. Then \begin{align*} \|\mathbb E[X\mid\mathcal G]\|_{L^p}\le\|X\|_{L^p}. \end{align*}

Description

Conditional expectation is a contraction on Lp spaces, so conditioning cannot increase the Lp norm of an integrable random variable.

Probability & Statistics

Extreme Information Levels

#4903

Probability & Statistics

Statement

Let $X\in L^1(\Omega,\mathcal F,\mathbb P)$. If $\mathcal G=\{\varnothing,\Omega\}$, then \begin{align*} \mathbb E[X\mid\mathcal G]=\mathbb E[X] \end{align*} $\mathbb P$-a.e. If $X$ is $\mathcal G$-measurable, then \begin{align*} \mathbb E[X\mid\mathcal G]=X \end{align*} $\mathbb P$-a.e.

Description

Conditional expectation reduces to the ordinary expectation under trivial information and to the variable itself when it is already measurable.

Probability & Statistics

Unit Ball Criterion for Compact Linear Operators

#4890

Analysis

Statement

Let $X$ and $Y$ be normed vector spaces over the same field $\mathbb{F} \in \{\mathbb{R}, \mathbb{C}\}$, and let $T: X \to Y$ be a [linear map](/page/Linear%20Map). Define the closed unit ball of $X$ by \begin{align*} B_X = \{x \in X : \|x\|_X \leq 1\}. \end{align*} Then $T$ is a compact linear operator if and only if $T(B_X)$ is relatively compact in $Y$, meaning that $\overline{T(B_X)}$ is compact in $Y$.

Description

This theorem reduces compactness of a linear operator to checking the image of the closed unit ball. It is a basic criterion in functional analysis because bounded sets can be scaled into the unit ball.

Analysis

Nonzero Spectrum of a Compact Operator

#4897

Analysis

Statement

Let $X$ be an infinite-dimensional complex [Banach space](/page/Banach%20Space) and let $T \in \mathcal{L}(X)$ be compact. If $\lambda \in \sigma(T)$ and $\lambda \ne 0$, then $\lambda$ is an eigenvalue of $T$. Moreover, the eigenspace $\ker(T-\lambda I)$ is finite-dimensional.

Description

This theorem describes the nonzero spectrum of a compact operator on an infinite-dimensional complex Banach space. Every nonzero spectral value is an eigenvalue with finite-dimensional eigenspace.

Analysis

Zero Belongs to the Spectrum in Infinite Dimension

#4898

Analysis

Statement

Let $X$ be an infinite-dimensional [Banach space](/page/Banach%20Space) and let $T \in \mathcal{L}(X)$ be compact. Then $0 \in \sigma(T)$.

Description

This theorem states that zero must belong to the spectrum of a compact operator on an infinite-dimensional Banach space. It reflects the impossibility of an invertible compact operator on such a space.

Analysis

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Theorems

Changing a Function on a Null Set

Linearity and Positivity

Doob-Dynkin Representation

Theorem #4905

Theorem #4907

$L^p$ Contraction

Extreme Information Levels

Unit Ball Criterion for Compact Linear Operators

Nonzero Spectrum of a Compact Operator

Zero Belongs to the Spectrum in Infinite Dimension

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Theorems

Changing a Function on a Null Set

Linearity and Positivity

Doob-Dynkin Representation

Theorem #4905

Theorem #4907

$L^p$ Contraction

Extreme Information Levels

Unit Ball Criterion for Compact Linear Operators

Nonzero Spectrum of a Compact Operator

Zero Belongs to the Spectrum in Infinite Dimension