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Let $X$ be a polish space equiped with the borel sigma-algebra and a probability measure $\mu$. How can one show that the set of all borel measurable functions $f:X\rightarrow R $ ($R$ being the real numbers), where two a.e. equal functions are identified, equiped with the topology of convegence in measure is separable?

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    Thanks (at least 15 characters...)2011-08-19

2 Answers 2

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Here's an outline of an argument, and it should be easy to fill in the details:

  1. Note that $L_0(X)$ is a metric space e.g. with respect to the metric $\displaystyle d(f,g) = \int \frac{|f-g|}{1+|f-g|}$.
  2. Choose a countable base $\{A_n\}_{n \in \mathbb{N}}$ for the topology on $X$.

    • Every open set is equal to the union of elements in $\{A_n\}$.
    • For every measurable set $E$ there is a $G_\delta$-set $G$ such that $\mu(E \triangle G) = 0$, that is $[E] = [G]$ in $L_{0}(X)$.
  3. Show that a non-negative measurable function $f$ is a pointwise monotone limit of simple functions.
    Hint: Put $B_{k,n} = \{x\in X : 2^{-n} k \leq f(x) \lt 2^{-n}(k+1)\}$ and consider $f_n = 2^{-n} \sum\limits_{k=0}^{2^{2n}}k \cdot[B_{k,n}]$.
  4. Split a general measurable function into positive and negative parts.

Use these observations to build a countable dense set of $L_{0}(X)$.

For completeness and further properties of $L_0(X)$, I recommend Driver's notes on probability Section 12, especially Theorem 12.8 on page 179. (Thanks to Nate Eldredge from whom I learned about these notes).

Edit: In view of Byron's answer, note that Driver's notes contain various forms of the functional monotone class theorem in Part II, Section 8 on pages 111ff. Of course, the main point in both our answers is that there is a countable generating and separating set for the $\sigma$-algebra. The assumption that $X$ be Polish ensures that.

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    Sorry, mainly I meant to add the *x in X* in the definition of your set B. The other modification is a (possibly not so nice after all) nicety. Theo, really, proceed at your convenience, this is *your* post...2011-08-20
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An alternative is to use the Functional Monotone Class Theorem. Let $\cal A$ be a countable collection of sets that generates ${\cal B}(X)$, and put ${\cal K}=\{1_A: A\mbox{ is a finite intersection of }{\cal A}\mbox{ sets }\}.$

Let ${\cal K}^\prime$ be the (countable!) $\mathbb{Q}$-vector space generated by $\cal K$, and set ${\cal M}=\{h: k^\prime_n\to h \mbox{ in probability for some }k^\prime_n\in{\cal K}^\prime\}.$

Then $\cal K$ and $\cal M$ satisfy the conditions of the FMCT, and hence $\cal M$ includes all bounded ${\cal B}(X)$-measurable functions. A truncation argument now shows that any ${\cal B}(X)$-measurable function can be approximated in probability by a sequence in ${\cal K}^\prime$.

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    I just wanted to point out that various useful variants of the FMCT are contained in Driver's notes I linked to, see the edit to my post. The dense set $\mathcal{K}'$ is of course the one I had in mind in my answer.2011-08-21