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*}

The collection of Lebesgue measurable subsets of $\mathbb{R}^n$ is a $\sigma$-algebra containing all open subsets of $\mathbb{R}^n$. The measure $\mathcal{L}^n$ is countably additive on this $\sigma$-algebra. If $(E_k)_{k=1}^\infty$ are pairwise disjoint Lebesgue measurable sets, then \begin{align*} \mathcal{L}^n\left(\bigcup_{k=1}^\infty E_k\right) = \sum_{k=1}^\infty \mathcal{L}^n(E_k). \end{align*} Moreover, $\mathcal{L}^n(\varnothing)=0$ and $\mathcal{L}^n(E) \le \mathcal{L}^n(F)$ whenever $E \subset F$ are Lebesgue measurable.

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.

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.

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.

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*}

Let $X$ be a [Banach space](/page/Banach%20Space), let $Y$ be a [normed vector space](/page/Normed%20Vector%20Space), and let $T: X \to Y$ be compact. If $x_k \rightharpoonup x$ weakly in $X$, then \begin{align*} \|Tx_k - Tx\|_Y &\to 0. \end{align*}