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 $(\Omega,\mathcal F,\mathbb P)$ 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 $(\Omega,\mathcal F)$ 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 $U\subset\mathbb{R}^n$ 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 $u,v\in C^2(U)\cap C(\overline{U})$ satisfy \begin{align*} -\Delta u\le -\Delta v \quad \text{in } U, \end{align*} and \begin{align*} u\le v \quad \text{on } \partial U. \end{align*} Equivalently, $-\Delta(u-v)\le 0$ in $U$ and $u-v\le 0$ on $\partial U$. Then $u\le v$ in $U$.

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*}

Let $U\subset\mathbb{R}^n$ be bounded and open. Let $A_0:H_0^1(U)\to H^{-1}(U)$ be the operator associated to a bounded and coercive uniformly elliptic principal [bilinear form](/page/Bilinear%20Form), so that $A_0$ is an isomorphism by Lax-Milgram. Let $K:H_0^1(U)\to H^{-1}(U)$ be a bounded operator such that \begin{align*} T=A_0^{-1}K:H_0^1(U)&\to H_0^1(U) \end{align*} is compact. Set $L=A_0+K$. Equip $H_0^1(U)$ with its usual Hilbert [inner product](/page/Inner%20Product), and let $(I+T)^*$ denote the adjoint of $I+T$ with respect to that inner product. Then the homogeneous solution space $\ker L\subset H_0^1(U)$ is finite-dimensional. If $\ker L=\{0\}$, then for every $f\in H^{-1}(U)$ there is a unique weak solution $u\in H_0^1(U)$ of $Lu=f$. If $\ker L\ne\{0\}$, then $Lu=f$ is…

Let $U\subset\mathbb{R}^n$ be open, and let $V\subsetneq W\subsetneq U$ be open sets with $\overline V$ compact and contained in $W$ and $\overline W$ compact and contained in $U$. Let $u\in H^1(U)$ satisfy \begin{align*} -\Delta u=f \end{align*} in the weak sense on $U$, where $f\in L^2(U)$. Then $u\in H^2(V)$ and there exists $C>0$, depending on $n$ and on the cutoff geometry needed to localize from $W$ to $V$, in particular on separation distances such as $\operatorname{dist}(V,\partial W)$, such that \begin{align*} \|u\|_{H^2(V)}\le C\left(\|f\|_{L^2(W)}+\|u\|_{L^2(W)}\right). \end{align*}

Let $U\subset\mathbb{R}^n$ be open, and let $A\in L^\infty(U;\mathbb{R}^{n\times n})$ be a real symmetric matrix field uniformly elliptic with constants $\theta$ and $\Theta$. Let $u\in H^1_{\mathrm{loc}}(U)$ be a weak solution of \begin{align*} -\sum_{i,j=1}^n \partial_{x_i}\left(A_{ij}\partial_{x_j}u\right)=0 \end{align*} in $U$. If $V\Subset W\Subset U$ are open sets, then there exist $\gamma\in(0,1)$ and $C>0$, depending on $n$, $\theta$, $\Theta$, and the compact-containment geometry of $V\Subset W$, such that $u$ has a representative in $C^{0,\gamma}(V)$ and \begin{align*} \|u\|_{L^\infty(V)} + \sup_{\substack{x,y\in V\\x\ne y}}\frac{|u(x)-u(y)|}{|x-y|^\gamma} \le C\|u\|_{L^2(W)}. \end{align*}