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?
How can one show that the topology of convergence in measure is separable?
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0Thanks (at least 15 characters...) – 2011-08-19
2 Answers
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.
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0Sorry, 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
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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0I 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