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 $H$ and $K$ be Hilbert spaces, let $U:H\to K$ be unitary, and let $E\subset H$ be an [orthonormal basis](/page/Orthonormal%20Basis) of $H$. Then $U(E)=\{Ue:e\in E\}$ is an orthonormal basis of $K$.

Let $H$ be a [Hilbert space](/page/Hilbert%20Space), let $E \subset H$ be an orthonormal set, and let $x \in H$. The set \begin{align*} \{e \in E : (x,e)_H \ne 0\} \end{align*} is countable.

Let $H$ be a [Hilbert space](/page/Hilbert%20Space) and let $E \subset H$ be an orthonormal set. The following conditions are equivalent. 1. $E$ is complete. 2. $\overline{\operatorname{span}}(E)=H$. 3. For every $x \in H$, \begin{align*} \|x\|_H^2 = \sup_{F \subset E,\ F \text{ finite}} \sum_{e \in F}|(x,e)_H|^2. \end{align*} 4. For every $x \in H$, the net of finite partial sums \begin{align*} \sum_{e \in F}(x,e)_H e, \end{align*} indexed by finite subsets $F \subset E$, converges to $x$ in $H$.

Let $H$ be a [Hilbert space](/page/Hilbert%20Space). If $E$ and $F$ are orthonormal bases of $H$, then $E$ and $F$ have the same cardinality.

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*}

Let $X$ be a set and let $\mathcal A$ be an algebra of sets on $X$. The smallest monotone class containing $\mathcal A$ is $\sigma(\mathcal A)$.

Let $(X,d)$ be a separable [metric space](/page/Metric%20Space). Then $\mathcal B(X)$ is countably generated.

Let $X$ be a set and let $\mathcal F$ be a finite sigma-algebra on $X$. Then the atoms of $\mathcal F$ form a finite partition of $X$, and every set in $\mathcal F$ is a union of atoms. Conversely, every finite partition of $X$ generates a finite sigma-algebra consisting exactly of unions of the partition elements.