Let $X$ be a continuum, meaning a nonempty compact connected Hausdorff [topological space](/page/Topological%20Space), and let $f: X \to \mathbb{R}$ be continuous, where $\mathbb{R}$ has its usual topology. Then there exist [real numbers](/page/Real%20Numbers) $a,b \in \mathbb{R}$ with $a \leq b$ such that \begin{align*} f(X) = [a,b]. \end{align*}

Let $(\Omega, \mathcal F, \mathbb P)$ be a probability space, and let $A, B \in \mathcal F$. If $\mathbb P(B) > 0$, then \begin{align*} \mathbb P(A \cap B) = \mathbb P(A \mid B)\mathbb P(B). \end{align*} If $\mathbb P(A) > 0$, then \begin{align*} \mathbb P(A \cap B) = \mathbb P(B \mid A)\mathbb P(A). \end{align*}

Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space, let $n \in \mathbb{N}$, and let $A_1,\dots,A_n \in \mathcal{F}$. For each $1 \le k \le n$, define the event \begin{align*} B_k := \bigcap_{j=1}^{k} A_j. \end{align*} Assume that $\mathbb{P}(B_k) > 0$ for every $1 \le k \le n-1$. Then \begin{align*} \mathbb{P}(B_n) = \mathbb{P}(A_1)\prod_{k=2}^{n} \mathbb{P}(A_k \mid B_{k-1}). \end{align*} Equivalently, \begin{align*} \mathbb P(A_1 \cap \cdots \cap A_n) &= \mathbb P(A_1)\mathbb P(A_2 \mid A_1)\mathbb P(A_3 \mid A_1 \cap A_2) \cdots \\ &\quad \cdot \mathbb P(A_n \mid A_1 \cap \cdots \cap A_{n-1}). \end{align*} For $n=1$, the product over $2 \le k \le n$ is interpreted as the empty product, equal to $1$.

Let $\Omega \subset \mathbb C$ be open, let $u \in C^2(\Omega;\mathbb R)$ satisfy $\Delta u = 0$ on $\Omega$, and let $z_0 \in \Omega$. Then there exist an open neighbourhood $U \subset \Omega$ of $z_0$ and a function $v: U \to \mathbb R$ such that the function $F: U \to \mathbb C$ defined by $F(z)=u(z)+iv(z)$ is holomorphic on $U$.

Let $\Omega \subset \mathbb C$ be open, let \begin{align*} f: \Omega &\to \mathbb C \end{align*} be a function whose real and imaginary parts \begin{align*} u: \Omega &\to \mathbb R, & v: \Omega &\to \mathbb R \end{align*} have first partial derivatives at $z_0 = x_0 + i y_0 \in \Omega$, and write $f = u + iv$. Define the Wirtinger derivatives at $z_0$ by \begin{align*} \partial_z f(z_0) &= \frac{1}{2}\left(\partial_x f(z_0) - i\partial_y f(z_0)\right), \\ \partial_{\bar z} f(z_0) &= \frac{1}{2}\left(\partial_x f(z_0) + i\partial_y f(z_0)\right). \end{align*} Then $f$ satisfies the [Cauchy-Riemann equations](/page/Cauchy-Riemann%20Equations) at $z_0$ if and only if \begin{align*} \partial_{\bar z} f(z_0)=0. \end{align*} When these equations hold at $z_0$, one has \begin{align*} \partial_z f(z_0)=\partial_x u(z_0)+i\partial_x v(z_0). \end{align*}…

Identify $\mathbb{C}$ with $\mathbb{R}^2$ by sending $x+iy$ to $(x,y)$. Let \begin{align*} A: \mathbb{R}^2 \to \mathbb{R}^2 \end{align*} be a real [linear map](/page/Linear%20Map) whose matrix with respect to the standard basis of $\mathbb{R}^2$ is \begin{align*} \begin{pmatrix} a & c \\ b & d \end{pmatrix}, \end{align*} where $a,b,c,d \in \mathbb{R}$. Then $A$ is complex-linear if and only if $a=d$ and $c=-b$. In that case, under the identification $\mathbb{C}\cong\mathbb{R}^2$, the map $A$ is multiplication by the complex number $a+ib$.

Let $(\Omega,\mathcal F,\mathbb P)$ be a probability space, and let $X:(\Omega,\mathcal F)\to(\mathbb R,\mathcal B(\mathbb R))$ be a real-valued random variable. Let $\mathcal L^1$ denote one-dimensional Lebesgue measure on $(\mathbb R,\mathcal B(\mathbb R))$. Then $X$ is absolutely continuous, meaning that its law $\mu_X:=\mathbb P\circ X^{-1}$ satisfies $\mu_X\ll\mathcal L^1$, if and only if there exists a nonnegative Borel measurable function $f_X:\mathbb R\to[0,\infty]$ such that \begin{align*} \mathbb P(X\in A)=\int_A f_X(x)\,d\mathcal L^1(x) \end{align*} for every $A\in\mathcal B(\mathbb R)$. The function $f_X$ is unique up to equality $\mathcal L^1$-a.e.

Let $(\Omega,\mathcal F,\mathbb P)$ be a probability space, let $(E,\mathcal E)$ be a measurable space, and let $X:(\Omega,\mathcal F)\to(E,\mathcal E)$ be a [random variable](/page/Random%20Variable), meaning an $\mathcal F/\mathcal E$-measurable map. Define the distribution, or pushforward measure, $\mu_X:\mathcal E\to[0,1]$ by \begin{align*} \mu_X(A)=\mathbb P(X^{-1}(A)) \end{align*} for every $A\in\mathcal E$, where $X^{-1}(A)=\{\omega\in\Omega:X(\omega)\in A\}$. Then for every nonnegative $\mathcal E$-measurable function $g:E\to[0,\infty]$, \begin{align*} \int_\Omega g(X(\omega))\,d\mathbb P(\omega)=\int_E g(x)\,d\mu_X(x). \end{align*}

Let $F: \mathbb{R} \to [0,1]$ be a function. There exists a probability space $(\Omega,\mathcal{F},\mathbb{P})$ and a real-valued [random variable](/page/Random%20Variable) $X:(\Omega,\mathcal{F}) \to (\mathbb{R},\mathcal{B}(\mathbb{R}))$ such that \begin{align*} F(x) = \mathbb{P}(X \leq x) \end{align*} for every $x \in \mathbb{R}$ if and only if $F$ is nondecreasing, right-continuous, and satisfies \begin{align*} \lim_{x \to -\infty} F(x) &= 0, & \lim_{x \to \infty} F(x) &= 1. \end{align*}

Let $(\Omega,\mathcal F,\mathbb P)$ be a probability space, let $\mathcal B(\mathbb R)$ denote the Borel $\sigma$-algebra on $\mathbb R$, and let $X:(\Omega,\mathcal F)\to([0,\infty),\mathcal B(\mathbb R)|_{[0,\infty)})$ be a nonnegative real-valued random variable. Then, as an identity in $[0,\infty]$, \begin{align*} \mathbb E[X]=\int_0^\infty \mathbb P(\{\omega\in\Omega:X(\omega)>t\})\,d\mathcal L^1(t). \end{align*}