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

Throughout, the field and order laws of $\mathbb{R}$ are used freely (The Structure of ℝ as a Field and an Ordered Ring (i), The Structure of ℝ as a Field and an Ordered Ring (ii), Rules of Arithmetic and Order in an Ordered Field (ii), Rules of Arithmetic and Order in an Ordered Field (iv), Rules of Arithmetic and Order in an Ordered Field (v), Rules of Arithmetic and Order in an Ordered Field (vi), Rules of Arithmetic and Order in an Ordered Field (x)). The elements of $\mathbb{E}[X\mid\mathcal{G}]$ are the $\mathcal{G}/\mathcal{B}(\mathbb{R})$-measurable elements of $\mathcal{L}^{1}(P)$ with the same integrals as $X$ over all sets of $\mathcal{G}$ (Conditional Expectation (ii), Conditional Expectations: Notation (ii)), and for $f\in\mathcal{L}^{1}(P)$ and $A\in\mathcal{F}$, $\int_{A}f\,dP=\mathbb{E}[f\mathbf{1}_{A}]$ (Integrals over Measurable Subsets (vi), Integrals over Measurable Subsets (ii)). For $\mathcal{G}/\mathcal{B}(\mathbb{R})$-measurable $S,T$ and $c\in\mathbb{R}$, the functions $S\pm T$, $cS$, $ST$ and $|S|$ are $\mathcal{G}/\mathcal{B}(\mathbb{R})$-measurable and the sets $\{S>T\}$, $\{S\ge c\}$ and $\{|S|\le c\}$ lie in $\mathcal{G}$ (Arithmetic of Measurable Real- and [0,∞]-Valued Maps (i), Measurable Maps: Generators, Composition, Continuous Maps, Restrictions, and Criteria for Real- and [0,∞]-Valued Maps (v), on $(\Omega,\mathcal{G})$). For nonnegative elements of $\mathcal{L}^{1}(P)$, the integrals as elements of $\mathcal{M}^{+}$ and of $\mathcal{L}^{1}(P)$ agree (Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion (vi)).

Rational points of $I$. Let $Q_{I}=I\cap\mathbb{Q}$, the rationals read in $\mathbb{R}$ (The Real Numbers, with the Natural Numbers, Integers and Rationals Identified with Subsets of the Reals, and Completeness (v)); it is countable (Countable Sets: a Bijection from ℕ onto ℕ×ℕ, Finite Sets, Subsets, Surjective Images, Images, Products, and the Integers and Rationals (ix), Countable Sets: a Bijection from ℕ onto ℕ×ℕ, Finite Sets, Subsets, Surjective Images, Images, Products, and the Integers and Rationals (iv)).

Rational points on both sides. For every $x\in I$ there are $q,q'\in Q_{I}$ with $q<x<q'$. Indeed, as $I$ is open there is a real $\varepsilon>0$ such that every $t$ with $|t-x|<\varepsilon$ lies in $I$ (Open Subset of a Metric Space (i), Metric Spaces: Balls, the Metric Topology, and the Real and Complex Numbers as Metric Spaces (iv)), and by density of the rationals (The Real Numbers, with the Natural Numbers, Integers and Rationals Identified with Subsets of the Reals, and Completeness (x)) there are rationals $q\in(x-\varepsilon,x)$ and $q'\in(x,x+\varepsilon)$. Since $P(\Omega)=1$, $\Omega$ has a point $\omega_{0}$, and $X(\omega_{0})\in I$; hence $Q_{I}\neq\emptyset$. So for every family $(A_{q})_{q\in Q_{I}}$ of sets in $\mathcal{G}$, the union and the intersection lie in $\mathcal{G}$ (Unions and Intersections over a Countable Index Set in a σ-Algebra (i), Unions and Intersections over a Countable Index Set in a σ-Algebra (ii)); and if $P(A_{q})=0$ for every $q$, then, taking a surjection $g:\mathbb{N}\to Q_{I}$ (Countable Set (i)), $\bigcup_{q}A_{q}=\bigcup_{m\in\mathbb{N}}A_{g(m)}$ has probability $0$ (Null Sets and Properties Holding Almost Everywhere (iv)).

Proof of (i). Let

$$G_{-}=\bigcap_{q\in Q_{I}}\{Y<q\},\qquad G_{+}=\bigcap_{q\in Q_{I}}\{Y>q\},$$

both in $\mathcal{G}$. If $\omega\notin G_{-}\cup G_{+}$, there are $q,q'\in Q_{I}$ with $q\le Y(\omega)\le q'$, so $Y(\omega)\in I$ (Intervals of the Real Line (i)); thus $\{Y\notin I\}\subseteq G_{-}\cup G_{+}$. For $\omega\in G_{-}$ choose $q\in Q_{I}$ with $q<X(\omega)$; then $Y(\omega)<q<X(\omega)$. Hence $(X-Y)\mathbf{1}_{G_{-}}\ge0$, with $\{(X-Y)\mathbf{1}_{G_{-}}>0\}=G_{-}$, and

$$\mathbb{E}\bigl[(X-Y)\mathbf{1}_{G_{-}}\bigr]=\int_{G_{-}}X\,dP-\int_{G_{-}}Y\,dP=0$$

(Rules for Expectations, Variances and Covariances (iii)), so $P(G_{-})=0$ (The Lebesgue Integral and Null Sets (iv)). In the same way, with $(Y-X)\mathbf{1}_{G_{+}}$, $P(G_{+})=0$. The set $\{Y\in I\}=Y^{-1}(I)$ is an event, as $I$ is open and so Borel (Borel Sets of a Metric Space: Open and Closed Sets, Points, Open Sets of Subspaces, and Subspaces (i), Borel Sigma-Algebra of a Metric Space (ii)). Therefore $P(Y\notin I)\le P(G_{-})+P(G_{+})=0$ (Basic Properties of a Measure (iii), Basic Properties of a Measure (vii)) and $P(Y\in I)=1$ (Basic Properties of a Measure (v)).

Proof of (ii). Step 1: the rational supporting lines. For $q\in Q_{I}$ let $c_{q}$ be the real number of Convex Functions on an Open Interval: Monotone Slopes, Supporting Lines, Continuity, and the Supremum of Rational Supporting Lines (ii) at $q$, and $d_{q}=\varphi(q)-c_{q}q$. By Convex Functions on an Open Interval: Monotone Slopes, Supporting Lines, Continuity, and the Supremum of Rational Supporting Lines (iii),

$$\varphi\bigl(X(\omega)\bigr)\ge\varphi(q)+c_{q}\bigl(X(\omega)-q\bigr)=c_{q}X(\omega)+d_{q}\qquad(\omega\in\Omega).$$

The constant $d_{q}$ belongs to $\mathbb{E}[d_{q}\mid\mathcal{G}]$ (Rules for Conditional Expectations (i)), so $c_{q}Y+d_{q}\in\mathbb{E}[c_{q}X+d_{q}\mid\mathcal{G}]$ (Rules for Conditional Expectations (iii)), and monotonicity (Rules for Conditional Expectations (iv)) gives $c_{q}Y+d_{q}\le Z$ a.s. Thus the set $B_{q}=\{c_{q}Y+d_{q}>Z\}\in\mathcal{G}$ is null, so $P(B_{q})=0$ (Null Sets and Properties Holding Almost Everywhere (i)), and $B=\bigcup_{q\in Q_{I}}B_{q}$ is an event with $P(B)=0$.

Step 2: conclusion. Let $\omega\notin B$ with $Y(\omega)\in I$. Then $Z(\omega)\ge\varphi(q)+c_{q}\bigl(Y(\omega)-q\bigr)$ for every $q\in Q_{I}$, so $Z(\omega)$ is an upper bound of the set whose supremum is $\varphi(Y(\omega))$ by Convex Functions on an Open Interval: Monotone Slopes, Supporting Lines, Continuity, and the Supremum of Rational Supporting Lines (v); hence $\varphi(Y(\omega))\le Z(\omega)$. Every $\omega$ outside the event $B$ of probability $0$ thus has $Y(\omega)\notin I$ or $\varphi(Y(\omega))\le Z(\omega)$, which is the assertion (A Property Holding Almost Everywhere (i), Null Set of a Measure (i)).

Proof of (iii). $X$ is integrable (Lᵖ Inclusions and Constants on a Finite Measure Space (i)), and $Y$ is measurable. Let $G=\{Y\ge0\}\in\mathcal{G}$.

The case $p=1$. Here $|f|^{1}=|f|$ (Properties of Real Powers (i)). Since $\Omega\setminus G\in\mathcal{G}$, $|Y|=Y\mathbf{1}_{G}-Y\mathbf{1}_{\Omega\setminus G}$ and $X\mathbf{1}_{G}-X\mathbf{1}_{\Omega\setminus G}\le|X|$,

$$\mathbb{E}|Y|=\int_{G}Y\,dP-\int_{\Omega\setminus G}Y\,dP=\int_{G}X\,dP-\int_{\Omega\setminus G}X\,dP=\mathbb{E}\bigl[X\mathbf{1}_{G}-X\mathbf{1}_{\Omega\setminus G}\bigr]\le\mathbb{E}|X|$$

(Rules for Expectations, Variances and Covariances (iii), Rules for Expectations, Variances and Covariances (iv), Rules for Expectations, Variances and Covariances (v)); in particular $Y\in\mathcal{L}^{1}(P)$.

The case $p>1$. Let $r=p/(p-1)$, so that $r>1$, $1/p+1/r=1$ and $(p-1)r=p$. For $n\in\mathbb{N}$ let $A_{n}=\{|Y|\le n\}\in\mathcal{G}$ and

$$W_{n}=|Y|^{p-1}\bigl(\mathbf{1}_{A_{n}\cap G}-\mathbf{1}_{A_{n}\setminus G}\bigr),$$

which is $\mathcal{G}/\mathcal{B}(\mathbb{R})$-measurable (Real Powers of the Absolute Value of a Measurable Function Are Measurable (i) on $(\Omega,\mathcal{G})$, Properties of Simple Functions and of the Simple Integral (i)). By the exponent laws for real powers (Properties of Real Powers (i), Properties of Real Powers (iv), Real Powers (ii)), using $Y=|Y|$ on $G$ and $Y=-|Y|$ off $G$,

$$|W_{n}|=|Y|^{p-1}\mathbf{1}_{A_{n}}\le n^{p-1},\qquad W_{n}Y=|Y|^{p}\mathbf{1}_{A_{n}},\qquad |W_{n}|^{r}=|Y|^{p}\mathbf{1}_{A_{n}},$$

the bound by Properties of Real Powers (vi). Let $a_{n}=\mathbb{E}\bigl[|Y|^{p}\mathbf{1}_{A_{n}}\bigr]\ge0$. As $W_{n}$ is bounded, Rules for Conditional Expectations (viii) and Rules for Conditional Expectations (ii) give

$$a_{n}=\mathbb{E}[W_{n}Y]=\mathbb{E}[W_{n}X].$$

The display shows $W_{n}\in\mathcal{L}^{r}(P)$ with $\|W_{n}\|_{r}=a_{n}^{1/r}$ (Power-Integrable Functions and the p-Seminorm (ii), Power-Integrable Functions and the p-Seminorm (iii)), so Hölder's inequality (Hölder's Inequality (i)) and Rules for Expectations, Variances and Covariances (v) give

$$a_{n}\le\mathbb{E}|W_{n}X|\le\|X\|_{p}\,a_{n}^{1/r}.$$

If $a_{n}>0$, then $a_{n}=a_{n}^{1/p}a_{n}^{1/r}$ with $a_{n}^{1/r}>0$ (Properties of Real Powers (iv), Properties of Real Powers (ii)), so $a_{n}^{1/p}\le\|X\|_{p}$ and, raising to the power $p$, $a_{n}\le\|X\|_{p}^{p}=\mathbb{E}|X|^{p}$ (Properties of Real Powers (vi), Properties of Real Powers (vii)); if $a_{n}=0$ this is clear. Finally $A_{n}\subseteq A_{n+1}$, and every $\omega$ lies in $A_{n}$ for $n>|Y(\omega)|$ (The Real Numbers, with the Natural Numbers, Integers and Rationals Identified with Subsets of the Reals, and Completeness (ix)), so the functions $|Y|^{p}\mathbf{1}_{A_{n}}$ increase pointwise to $|Y|^{p}$, and monotone convergence (Monotone Convergence Theorem (ii)) gives

$$\mathbb{E}|Y|^{p}=\sup_{n}a_{n}\le\mathbb{E}|X|^{p}<\infty,$$

so $Y\in\mathcal{L}^{p}(P)$ (Power-Integrable Functions and the p-Seminorm (ii)).

Proof of (iv). Step 1: ordering the versions. Let $\omega$ be such that $X_{j}(\omega)\to X(\omega)$, and $k\in\mathbb{N}$. As $X_{k}(\omega)\le X_{j}(\omega)$ for every $j\ge k$, Limits of Sequences of Real Numbers: Uniqueness, Boundedness, Constants, Tails, Arithmetic, Quotients, Absolute Values, Finite Sums, Order, Squeezing, Domination and Subsequences (x) gives $X_{k}(\omega)\le X(\omega)$. Hence $X_{k}\le X$ a.s. (Null Sets and Properties Holding Almost Everywhere (vi)), and $X_{k}\le X_{k+1}$ everywhere, so by monotonicity (Rules for Conditional Expectations (iv)) $Y_{k}\le Y_{k+1}$ a.s. and $Y_{k}\le Y$ a.s. The sets $C_{k}=\{Y_{k}>Y_{k+1}\}\cup\{Y_{k}>Y\}\in\mathcal{G}$ are therefore null, so $P(C_{k})=0$ (Null Sets and Properties Holding Almost Everywhere (iii), Null Sets and Properties Holding Almost Everywhere (i)), and the event $A=\Omega\setminus\bigcup_{k\in\mathbb{N}}C_{k}$ has $P(\Omega\setminus A)=0$ (Null Sets and Properties Holding Almost Everywhere (iv)).

Step 2: the expectations. Since $0\le X_{k}\le X$ a.s. and $X_{k}\to X$ a.s., $\mathbb{E}X_{k}\to\mathbb{E}X$ (The Lebesgue Integral and Null Sets (ix), with dominating function $X$). Let $h_{k}=(Y-Y_{k})\mathbf{1}_{A}$, a measurable function with $h_{k}\ge0$, equal to $Y-Y_{k}$ a.s. By The Lebesgue Integral and Null Sets (iii), Rules for Expectations, Variances and Covariances (iii) and Rules for Conditional Expectations (ii),

$$\mathbb{E}h_{k}=\mathbb{E}Y-\mathbb{E}Y_{k}=\mathbb{E}X-\mathbb{E}X_{k}\to0$$

(Limits of Sequences of Real Numbers: Uniqueness, Boundedness, Constants, Tails, Arithmetic, Quotients, Absolute Values, Finite Sums, Order, Squeezing, Domination and Subsequences (i), Limits of Sequences of Real Numbers: Uniqueness, Boundedness, Constants, Tails, Arithmetic, Quotients, Absolute Values, Finite Sums, Order, Squeezing, Domination and Subsequences (vi)).

Step 3: the limit. Let $h(\omega)=\inf_{k}h_{k}(\omega)$, a measurable function into $[0,\infty]$ (Measurable Maps: Generators, Composition, Continuous Maps, Restrictions, and Criteria for Real- and [0,∞]-Valued Maps (vii), Suprema, Infima, Limits Inferior and Limits Superior of Measurable [0,∞]-Valued Maps (ii)) with $h\le h_{k}$ for every $k$. Hence $\int h\,dP\le\mathbb{E}h_{k}$ for every $k$ (Basic Properties of the Integral of Nonnegative Measurable Maps (iii)), so $\int h\,dP$ is a real number $\le0$ (Limits of Sequences of Real Numbers: Uniqueness, Boundedness, Constants, Tails, Arithmetic, Quotients, Absolute Values, Finite Sums, Order, Squeezing, Domination and Subsequences (x)), that is, $\int h\,dP=0$, and $P(h>0)=0$ (The Lebesgue Integral and Null Sets (iv)). Let $\omega\in A$ with $h(\omega)=0$. The sequence $(h_{k}(\omega))_{k}=(Y(\omega)-Y_{k}(\omega))_{k}$ is nonincreasing and bounded below by $0$, so it converges to its infimum $h(\omega)=0$ (Completeness of the Real Numbers for Sequences: Monotone Convergence, the Bolzano-Weierstrass Theorem and Cauchy Sequences (iii)), and $Y_{k}(\omega)\to Y(\omega)$ (Limits of Sequences of Real Numbers: Uniqueness, Boundedness, Constants, Tails, Arithmetic, Quotients, Absolute Values, Finite Sums, Order, Squeezing, Domination and Subsequences (vi)). So $Y_{k}(\omega)\to Y(\omega)$ for every $\omega$ outside the event $(\Omega\setminus A)\cup\{h>0\}$, which has probability $0$ (Basic Properties of a Measure (vii)).

Proof of (v). By The Lebesgue Integral and Null Sets (vii), $X$ and every $X_{k}$ are integrable.

Proof of (vi). By (v) the conditional expectations of $X$ and the $X_{k}$ are defined, and by The Lebesgue Integral and Null Sets (viii), $\mathbb{E}|X_{k}-X|\to0$. By Rules for Conditional Expectations (iii), $Y_{k}-Y\in\mathbb{E}[X_{k}-X\mid\mathcal{G}]$, so (iii) with $p=1$ gives

$$0\le\mathbb{E}|Y_{k}-Y|\le\mathbb{E}|X_{k}-X|\qquad(k\in\mathbb{N}),$$

and $\mathbb{E}|Y_{k}-Y|\to0$ by squeezing (Limits of Sequences of Real Numbers: Uniqueness, Boundedness, Constants, Tails, Arithmetic, Quotients, Absolute Values, Finite Sums, Order, Squeezing, Domination and Subsequences (i), Limits of Sequences of Real Numbers: Uniqueness, Boundedness, Constants, Tails, Arithmetic, Quotients, Absolute Values, Finite Sums, Order, Squeezing, Domination and Subsequences (xi)).