Conditional Jensen Inequality, Lᵖ Contraction and Convergence Theorems for Conditional Expectations

In the setting of Conditional Expectations: Notation, the following hold for conditional expectations given $\mathcal{G}$.

In the next two clauses, let $I$ be an interval that is an open subset of $(\mathbb{R},d_{\mathbb{R}})$ (Metric Spaces: Balls, the Metric Topology, and the Real and Complex Numbers as Metric Spaces (iv)), let $X\in\mathcal{L}^{1}(P)$ with $X(\omega)\in I$ for every $\omega\in\Omega$, and let $Y\in\mathbb{E}[X\mid\mathcal{G}]$.

(i) Values in the interval. $P(Y\in I)=1$.

(ii) Conditional Jensen inequality. Let $\varphi:I\to\mathbb{R}$ be convex on $I$ with $\varphi(X)\in\mathcal{L}^{1}(P)$, and let $Z\in\mathbb{E}[\varphi(X)\mid\mathcal{G}]$. Then

$$\varphi\bigl(Y(\omega)\bigr)\le Z(\omega)\qquad\text{for almost every }\omega\text{ with }Y(\omega)\in I .$$

(iii) Contraction in $\mathcal{L}^{p}$. Let $p\ge1$ be a real number, let $X\in\mathcal{L}^{p}(P)$ (so that $X\in\mathcal{L}^{1}(P)$ by Lᵖ Inclusions and Constants on a Finite Measure Space (i)) and let $Y\in\mathbb{E}[X\mid\mathcal{G}]$. Then $Y\in\mathcal{L}^{p}(P)$ and

$$\mathbb{E}|Y|^{p}\le\mathbb{E}|X|^{p} .$$

(iv) Monotone convergence. Let $(X_{k})_{k\in\mathbb{N}}$ be a sequence in $\mathcal{L}^{1}(P)$ and $X\in\mathcal{L}^{1}(P)$ with $0\le X_{k}\le X_{k+1}$ for every $k\in\mathbb{N}$ and $X_{k}(\omega)\to X(\omega)$ for almost every $\omega$. Let $(Y_{k})_{k\in\mathbb{N}}$ be a sequence with $Y_{k}\in\mathbb{E}[X_{k}\mid\mathcal{G}]$ for every $k$, and let $Y\in\mathbb{E}[X\mid\mathcal{G}]$. Then $Y_{k}(\omega)\to Y(\omega)$ for almost every $\omega$.

In the next two clauses, let $(X_{k})_{k\in\mathbb{N}}$ be a sequence of random variables, $X$ a random variable and $V\in\mathcal{L}^{1}(P)$, with $X_{k}(\omega)\to X(\omega)$ for almost every $\omega$ and, for every $k\in\mathbb{N}$, $|X_{k}|\le V$ a.s.

(v) Integrability. $X\in\mathcal{L}^{1}(P)$ and $X_{k}\in\mathcal{L}^{1}(P)$ for every $k\in\mathbb{N}$.

(vi) Dominated convergence in mean. For every sequence $(Y_{k})_{k\in\mathbb{N}}$ with $Y_{k}\in\mathbb{E}[X_{k}\mid\mathcal{G}]$ for every $k$ and every $Y\in\mathbb{E}[X\mid\mathcal{G}]$,

$$\mathbb{E}|Y_{k}-Y|\to0 .$$