Let $U\subsetneq\mathbb R^n$ be bounded and open. Suppose $a_{ij},b_i,c\in L^\infty(U)$, and let $B:H_0^1(U)\times H_0^1(U)\to\mathbb R$ be the [bilinear form](/page/Bilinear%20Form) associated to the divergence form operator \begin{align*} Lu&=-\sum_{i,j=1}^n\partial_{x_i}(a_{ij}\partial_{x_j}u)+\sum_{i=1}^n b_i\partial_{x_i}u+c u. \end{align*} Then there exists $C>0$ such that \begin{align*} |B[u,v]|&\le C\|u\|_{H^1(U)}\|v\|_{H^1(U)} \end{align*} for every $u,v\in H_0^1(U)$.

Let $U\subsetneq\mathbb R^n$ be open and let $L$ be a second-order nondivergence form operator with real symmetric coefficient matrix $A=(a_{ij})$. Then $A$ is uniformly elliptic with constant $\theta>0$ iff \begin{align*} p_L(x,\xi)&\ge \theta |\xi|^2 \end{align*} for $\mathcal L^n$-a.e. $x\in U$ and every $\xi\in\mathbb R^n$.

Let $(\Omega,\mathcal F,\mathbb P)$ be a probability space. If $A_1,\dots,A_n\in\mathcal F$ are mutually independent, then for every choice $B_i\in\{A_i,A_i^c\}$, the events $B_1,\dots,B_n$ are mutually independent.

Let $(E_i,\mathcal E_i,\mu_i)$ be probability spaces for $i\in\{1,\dots,n\}$, and set \begin{align*} (E,\mathcal E,\mu)=(E_1\times\cdots\times E_n,\mathcal E_1\otimes\cdots\otimes\mathcal E_n,\mu_1\otimes\cdots\otimes\mu_n). \end{align*} Let $\pi_i:E\to E_i$ be the coordinate projection $\pi_i(x_1,\dots,x_n)=x_i$. Then $\pi_1,\dots,\pi_n$ are independent and $\mu_{\pi_i}=\mu_i$.

Let $(X_i)_{i\in I}$ be an independent family of random variables on $(\Omega,\mathcal F,\mathbb P)$. Let $I_1,\dots,I_m\subset I$ be pairwise disjoint, and set \begin{align*} \mathcal G_r=\sigma(X_i:i\in I_r), \qquad r\in\{1,\dots,m\}. \end{align*} Then $\mathcal G_1,\dots,\mathcal G_m$ are independent sigma-algebras.

Let $X_1,\dots,X_n$ be independent real-valued random variables. Then, for every $u\in\mathbb R$, \begin{align*} \phi_{X_1+\cdots+X_n}(u)=\prod_{i=1}^n\phi_{X_i}(u). \end{align*}

Let $(X,Z)$ be a real-valued and $\mathbb R^n$-valued pair of random variables, so that $X:\Omega\to\mathbb R$ and $Z:\Omega\to\mathbb R^n$. Suppose $(X,Z)$ has joint density $f_{X,Z}$ with respect to $\mathcal L^{n+1}$, let $f_Z$ be the marginal density of $Z$, and suppose $X\in L^1(\Omega,\mathcal F,\mathbb P)$. Using the measurable joint density, define the conditional mean by taking the positive and negative part integrals and subtracting them where both are finite: \begin{align*} g:\mathbb R^n&\to\mathbb R \\ z&\mapsto \begin{cases} \int_{\mathbb R} x\, f_{X\mid Z}(x\mid z)\,d\mathcal L^1(x), & f_Z(z)>0 \text{ and the integral is finite},\\ 0, & \text{otherwise}. \end{cases} \end{align*} If the defining integral is finite for $f_Z(z)\,d\mathcal L^n(z)$-a.e. $z$, then $g(Z)\in L^1(\Omega,\mathcal F,\mathbb P)$…

Let $(\Omega,\mathcal F,\mathbb P)$ be a probability space with filtration $(\mathcal F_t)_{t\in T}$, and let $(X_t)_{t\in T}$ be an integrable adapted process. Then $(X_t)_{t\in T}$ is a martingale if and only if for every $s\le t$, \begin{align*} \mathbb E[X_t-X_s\mid\mathcal F_s]=0 \end{align*} $\mathbb P$-a.s.

Let $(\Omega,\mathcal F,\mathbb P)$ be a probability space, let $X\in L^1(\Omega,\mathcal F,\mathbb P)$, and let $(A_i)_{i=1}^n$ be a finite information partition. Let $\mathcal G=\sigma(A_1,\dots,A_n)$. If $\mathbb P(A_i)>0$ for each $i$, then \begin{align*} \mathbb E[X\mid\mathcal G] =\sum_{i=1}^n \left(\frac{1}{\mathbb P(A_i)}\int_{A_i} X\,d\mathbb P\right)\mathbb 1_{A_i}. \end{align*}

Let $H$ be a [Hilbert space](/page/Hilbert%20Space) with [orthonormal basis](/page/Orthonormal%20Basis) $E$, and let $(\lambda_e)_{e\in E}$ be a bounded family of scalars. The formula \begin{align*} Tx = \sum_{e\in E}\lambda_e (x,e)_H e \end{align*} defines a [bounded linear operator](/page/Bounded%20Linear%20Operator) $T\in\mathcal L(H)$, and \begin{align*} \|T\|_{\mathcal L(H)} = \sup_{e\in E}|\lambda_e|. \end{align*}