Assuming the [axiom of choice](/page/Axiom%20of%20Choice), there exists a subset $V \subsetneq [0,1]$ that does not belong to the Lebesgue sigma-algebra on $\mathbb R$.

Let $(\Omega,\mathcal F)$ be a measurable space, and let $(E_i,\mathcal E_i)$ be measurable spaces for $i \in \{1,\dots,n\}$. A map \begin{align*} X: \Omega &\to E_1 \times \cdots \times E_n \end{align*} with coordinate maps $X_i: \Omega \to E_i$ is measurable from $(\Omega,\mathcal F)$ to $(E_1 \times \cdots \times E_n,\mathcal E_1 \otimes \cdots \otimes \mathcal E_n)$ iff each coordinate map $X_i$ is measurable.

Let $(\Omega,\mathcal F)$ be a measurable space and let $X: \Omega \to \mathbb R$ be a function. Then $X$ is measurable as a map from $(\Omega,\mathcal F)$ to $(\mathbb R,\mathcal B(\mathbb R))$ iff \begin{align*} \{\omega \in \Omega : X(\omega) \le a\} \in \mathcal F \end{align*} for every $a \in \mathbb R$.

Let $(\Omega,\mathcal F)$ be a measurable space and let $(A_n)_{n=1}^{\infty}$ be a sequence of events. Then \begin{align*} \limsup_{n \to \infty} A_n \in \mathcal F, \qquad \liminf_{n \to \infty} A_n \in \mathcal F. \end{align*}

Let $\mathcal A \subset \mathcal P(\Omega)$. Then $\sigma(\mathcal A)$ is a sigma-algebra on $\Omega$, contains $\mathcal A$, and satisfies the following minimality property: if $\mathcal G$ is any sigma-algebra on $\Omega$ with $\mathcal A \subset \mathcal G$, then $\sigma(\mathcal A) \subset \mathcal G$.

Let $\Omega$ be a set. For each $i \in I$, let $(E_i,\mathcal E_i)$ be a measurable space and let $X_i: \Omega \to E_i$ be a function. Then $\sigma(X_i : i \in I)$ is the smallest sigma-algebra $\mathcal F$ on $\Omega$ for which every map $X_i: (\Omega,\mathcal F) \to (E_i,\mathcal E_i)$ is measurable.

Let $(\Omega,\mathcal F,\mathbb P)$ be a probability space, and let $X_i: (\Omega,\mathcal F) \to (E_i,\mathcal E_i)$ be random variables for $i \in \{1,\dots,n\}$. Suppose each $\mathcal E_i$ is generated by a class $\mathcal A_i \subset \mathcal E_i$ such that $E_i \in \mathcal A_i$ and $\mathcal A_i$ is closed under finite intersections. Then $X_1,\dots,X_n$ are independent iff \begin{align*} \mathbb P\left(\bigcap_{i=1}^{n}\{\omega \in \Omega : X_i(\omega) \in A_i\}\right)=\prod_{i=1}^{n}\mathbb P(X_i^{-1}(A_i)) \end{align*} for every $A_i \in \mathcal A_i$.

Let $U\subset\mathbb{R}^n$ be bounded and open. Let $L$ be a divergence form [elliptic operator](/page/Elliptic%20Operator) with $b_i=0$ for $1\le i\le n$, with $c\ge 0$ $\mathcal{L}^n$-a.e. in $U$, and with leading coefficients satisfying the ellipticity bound with constant $\theta>0$. Let $B:H_0^1(U)\times H_0^1(U)\to\mathbb R$ denote the associated bilinear form, \begin{align*} B[u,v] &=\int_U \sum_{i,j=1}^n a_{ij}\partial_{x_j}u\,\partial_{x_i}v+cuv\,d\mathcal L^n . \end{align*} Then there exists $\alpha>0$, depending on $U$ and $\theta$, such that \begin{align*} B[u,u]\ge \alpha\|u\|_{H_0^1(U)}^2 \end{align*} for every $u\in H_0^1(U)$.

Let $U\subset\mathbb{R}^n$ be bounded and open. Let $A\in L^\infty(U;\mathbb{R}^{n\times n})$ be uniformly elliptic with constants $\theta$ and $\Theta$. Let $u\in H_0^1(U)$ satisfy \begin{align*} \int_U \sum_{i,j=1}^n A_{ij}\partial_{x_j}u\,\partial_{x_i}v\,d\mathcal{L}^n = f(v) \end{align*} for every $v\in H_0^1(U)$, where $f\in H^{-1}(U)$. Then there exists $C>0$, depending only on $U$ and $\theta$, such that \begin{align*} \|u\|_{H_0^1(U)}\le C\|f\|_{H^{-1}(U)}. \end{align*}

Let $U\subset\mathbb{R}^n$ be bounded and open. Let $L$ be a divergence form [elliptic operator](/page/Elliptic%20Operator) whose associated [bilinear form](/page/Bilinear%20Form) $B$ is bounded on $H_0^1(U)\times H_0^1(U)$ and coercive on $H_0^1(U)$. Then for every $f\in H^{-1}(U)$ there exists a unique $u\in H_0^1(U)$ such that \begin{align*} B[u,v]=f(v) \end{align*} for every $v\in H_0^1(U)$. Moreover, there exists $C>0$ such that \begin{align*} \|u\|_{H_0^1(U)}\le C\|f\|_{H^{-1}(U)}. \end{align*}