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 $\mathcal F$.