FoldUnfold Table of Contents The Monotone Convergence Theorem for Nonnegative Measurable Functions The Monotone Convergence Theorem for Nonnegative Measurable Functions The Monotone Convergence Theorem for Series of Nonnegative Measurable Functions The Monotone Convergence Theorem for Nonnegative Measurable Functions The Monotone Convergence Theorem for Nonnegative … [Read more...]

## Fatou’s Lemma for Nonnegative Measurable Functions

FoldUnfold Table of Contents Fatou's Lemma for Nonnegative Measurable Functions Fatou's Lemma for Nonnegative Measurable Functions Recall from the Fatou's Lemma for Nonnegative Lebesgue Measurable Functions page that if $(f_n(x))_{n=1}^{\infty}$ is a sequence of nonnegative Lebesgue measurable functions defined on a Lebesgue measurable set $E$ such that: 1) … [Read more...]

## Chebyshev’s Inequality for Nonnegative Measurable Functions

FoldUnfold Table of Contents Chebyshev's Inequality for Nonnegative Measurable Functions Chebyshev's Inequality for Nonnegative Measurable Functions Theorem 1 (Chebyshev's Inequality for Nonnegative Measurable Functions): Let $(X, \mathcal A, \mu)$ be a measure space and let $f$ be a nonnegative measurable function defined on $X$. Then for all $\lambda \in \mathbb{R}$, … [Read more...]

## The Integral of Nonnegative Measurable Functions

FoldUnfold Table of Contents The Integral of Nonnegative Measurable Functions The Integral of Nonnegative Measurable Functions Recall from The Integral of Nonnegative Simple Functions page that if $(X, \mathcal A, \mu)$ is a complete measure space then the integral of a nonnegative simple function $\varphi$ defined on a measurable set $E$ is as follows. For $\psi(x) = 0$ … [Read more...]

## The Integral of Nonnegative Simple Functions

FoldUnfold Table of Contents The Integral of Nonnegative Simple Functions The Integral of Nonnegative Simple Functions Recall from The Lebesgue Integral of Simple Functions page that if $\varphi$ is a simple function defined on a Lebesgue measurable set $E$ with $m(E) and with canonical representation: (1) \begin{align} \quad \varphi(x) = \sum_{k=1}^{n} a_k \chi_{E_k}(x) … [Read more...]

## The Simple Function Approximation Lemma and Theorem for General Measurable Spaces

FoldUnfold Table of Contents The Simple Function Approximation Lemma and Theorem for General Measurable Spaces The Simple Function Approximation Lemma for General Measurable Spaces The Simple Function Approximation Theorem for General Measurable Spaces The Simple Function Approximation Lemma and Theorem for General Measurable Spaces The Simple Function Approximation Lemma … [Read more...]

## Sums, Multiples, and Products of Measurable Functions

FoldUnfold Table of Contents Sums, Multiples, and Products of Measurable Functions Sums, Multiples, and Products of Measurable Functions Recall from the General Measurable Functions page that if $(X, \mathcal A)$ is a measurable space then an extended real-valued function $f$ defined on a measurable set $E$ is said to be measurable function on $E$ if for all $\alpha \in … [Read more...]

## General Measurable Functions

FoldUnfold Table of Contents General Measurable Functions General Measurable Functions Recall that an extended real-valued function $f$ is said to be Lebesgue measurable on its domain (assumed to be a Lebesgue measurable set) if for every $\alpha \in \mathbb{R}$ the set $\{ x \in D(f) : f(x) is a Lebesgue measurable set. Given any measurable space $(X, \mathcal A)$, we … [Read more...]

## The Borel-Cantelli Lemma

FoldUnfold Table of Contents The Borel-Cantelli Lemma The Borel-Cantelli Lemma Lemma (The Borel-Cantelli Lemma): Let $(X, \mathcal A, \mu)$ be a measure space and let $(E_n)_{n=1}^{\infty}$ be a collection of measurable sets such that $\displaystyle{\sum_{n=1}^{\infty} \mu (E_n) . Then almost every $x \in X$ belongs to at most a finite number of sets in … [Read more...]

## The Completion of a Measure Space

FoldUnfold Table of Contents The Completion of a Measure Space The Completion of a Measure Space Recall from the Complete Measure Spaces page that a measure space $(X, \mathcal, \mu)$ is said to be complete if for every measurable set $E \in \mathcal P(X)$ with $\mu(E) = 0$ we have that every subset of $E$ is a measurable set. Given an incomplete measure space $(X, … [Read more...]