Radon-Nikodym Theorem/Lemma 1

Lemma
Let $\struct {X, \Sigma}$ be a measurable space.

Let $\mu$ and $\nu$ be $\sigma$-finite measures on $\struct {X, \Sigma}$ such that:


 * $\nu$ is absolutely continuous with respect to $\mu$.

Define $\mathcal F$ to be the set of $\Sigma$-measurable functions $f : X \to \overline \R_{\ge 0}$ with:


 * $\ds \int_A f \rd \mu \le \map \nu A$

for each $A \in \Sigma$. Let $\sequence {f_n}_{n \mathop \in \N}$ be a sequence in $\mathcal F$.

Then:


 * for each $n \in \N$, the pointwise maximum $\max \set {f_1, f_2, \ldots, f_n}$ is contained in $\mathcal F$.

Proof
Let $A \in \Sigma$.

We first prove that for each $f, g \in \mathcal F$, we have:


 * $\max \set {f, g} \in \mathcal F$

From Measurable Functions Determine Measurable Sets, we have:


 * $\set {x \in X : \map f x < \map g x}$ is $\Sigma$-measurable

and:


 * $\set {x \in X : \map f x \ge \map g x}$ is $\Sigma$-measurable.

Define:


 * $A_1 = \set {x \in X : \map f x < \map g x} \cap A$

and:


 * $A_2 = \set {x \in X : \map f x \ge \map g x} \cap A$

For $x \in A_1$, we have:


 * $\map {\max \set {f, g} } x = \map g x$

and for $x \in A_1$, we have:


 * $\map {\max \set {f, g} } x = \map f x$

From Sigma-Algebra Closed under Countable Intersection, we have:


 * $A_1$ and $A_2$ are $\Sigma$-measurable.

Clearly we also have:


 * $A = A_1 \cup A_2$

with $A_1$ and $A_2$ disjoint.

From Pointwise Maximum of Measurable Functions is Measurable, we have:


 * $\max \set {f, g}$ is $\Sigma$-measurable.

Then:

So, we have:


 * $\ds \int_A \max \set {f, g} \rd \mu \le \map \nu A$

for each $A \in \Sigma$.

So:


 * $\max \set {f, g} \in \mathcal F$.

We now prove that:


 * for each $n \in \N$, the pointwise maximum $\max \set {f_1, f_2, \ldots, f_n}$ is contained in $\mathcal F$.

by induction.

For all $n \in \N$ let $\map P n$ be the proposition:


 * $\max \set {f_1, f_2, \ldots, f_n} \in \mathcal F$

Basis for Induction
We have:


 * $\max \set {f_1} = f_1$

so:


 * $\max \set {f_1} \in \mathcal F$

This is our basis for the induction.

Induction Hypothesis
Now we need to show that, if $\map P n$ is true, where $n \ge 1$, then it logically follows that $\map P {n + 1}$ is true.

Our induction hypothesis is:


 * $\max \set {f_1, f_2, \ldots, f_n} \in \mathcal F$

We aim to show that:


 * $\max \set {f_1, f_2, \ldots, f_{n + 1} } \in \mathcal F$

Induction Step
We have:


 * $f_{n + 1} \in \mathcal F$

and:


 * $\max \set {f_1, f_2, \ldots, f_n} \in \mathcal F$

so:

as required.

We therefore obtain:


 * for each $n \in \N$, the pointwise maximum $\max \set {f_1, f_2, \ldots, f_n}$ is contained in $\mathcal F$.