How can one show that the topology of convergence in measure is separable? 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?
 A: 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 ${\cal K}^\prime$.  
A: Here's an outline of an argument, and it should be easy to fill in the details:


*

*Note that $L_0(X)$ is a metric space e.g. with respect to the metric $\displaystyle d(f,g) = \int \frac{|f-g|}{1+|f-g|}$.

*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 $L_{0}(X)$.


*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 $f_n = 2^{-n} \sum\limits_{k=0}^{2^{2n}}k \cdot[B_{k,n}]$.

*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.
