Measurable Maps and Continuous Functions

Solution 1:

If $\Omega_1$ and $\Omega_2$ are topological spaces and $\mathcal{B}_1$ and $\mathcal{B}_2$ are their Borel sigma-algebras, then every continuous function is measurable (but, in general, not every measurable function is continuous).

Proof. Let $u:\Omega_1\to\Omega_2$ be a continuous function. That $u$ is $\mathcal{B}_1/\mathcal{B}_2$ measurable is equivalent to the assertion that $\mathcal{C}=\mathcal{B}_2$, where $$ \mathcal{C}=\{B\in\mathcal{B}_2\mid u^{-1}(B)\in\mathcal{B}_1\}. $$ Thus, one has to show that $\mathcal{C}=\mathcal{B}_2$. This is done in two steps.

• First, $u$ is continuous hence one already knows that $\mathcal{T}_2\subseteq\mathcal{C}$, where $\mathcal{T}_2$ is the topology of $\Omega_2$ (in other words, $\mathcal{T}_2$ is the collection of its open sets). Since $\mathcal{C}\subseteq\mathcal{B}_2$ by definition, one gets the double inclusion $$ \mathcal{T}_2\subseteq\mathcal{C}\subseteq\mathcal{B}_2. $$ By definition of Borel sigma-algebras, $\mathcal{B}_2=\sigma(\mathcal{T}_2)$, that is, $\mathcal{B}_2$ is the sigma-algebra generated by $\mathcal{T}_2$ (or equivalently, the smallest sigma-algebra which contains $\mathcal{T}_2$). Hence, the double inclusion above implies $$\mathcal{B}_2=\sigma(\mathcal{T}_2)\subseteq\sigma(\mathcal{C})\subseteq\sigma(\mathcal{B}_2)=\mathcal{B}_2, $$ that is, $\sigma(\mathcal{C})=\mathcal{B}_2$.
• Second, and this will be enough to conclude, one wants to show that $\mathcal{C}=\sigma(\mathcal{C})$, or equivalently, that $\mathcal{C}$ is a sigma-algebra. It happens that $\mathcal{C}$ fulfills all the axioms defining a sigma-algebra and that it is easy to show, so I will be lazy and leave you this part of the proof.

Solution 2:

(While the answer is not very useful to the OP, I am not deleting it since I do believe that someone will find it useful, and perhaps even the OP someday may find it useful.)

[This answer is using some amount of choice, in particular a countable union of countable sets is countable assumed in here]

The Borel measurable functions are constructed as iterated pointwise limits of continuous functions.

For this we need to elaborate a little bit on the construction of $\mathcal B$, that is the Borel $\sigma$-algebra.

• $\Sigma^0_0=\Pi^0_0=$ a countable basis of open intervals (or balls),
• For $\alpha$ a countable ordinal denote: $$ \Sigma^0_\alpha=\{\bigcup_{i\in\mathbb N} A_i\mid A_i\in\bigcup_{\beta<\alpha}\Pi^0_\beta\},\quad \Pi^0_\alpha = \{X\setminus A\mid A\in\Sigma^0_\alpha\},\quad \Delta^0_\alpha=\Sigma^0_\alpha\cap\Pi^0_\alpha $$

The $\Sigma^0_\alpha$ are closed under countable unions, the $\Pi^0_\alpha$ under countable intersections and $\Delta^0_\alpha$ are algebras (but not $\sigma$-algebra in general), and the whole hierarchy is closed under inclusion, that is: $\beta<\alpha$ then $\Sigma^0_\beta\subseteq\Delta^0_\alpha\subseteq\Sigma^0_\alpha$, and similarly for $\Pi^0_\alpha$.

We have that: $$\mathcal B=\bigcup_{\alpha<\aleph_1}\large\Delta^0_\alpha$$ (That is a union of all the countable levels of the construction). In the case of $\mathbb R^n$ (and in any case that the space has similar properties to these spaces) we cannot have this union any shorter. That is, on every stage there is a new set added to the construction.

We also have that $|\mathcal B|=|\mathbb R|=2^{\aleph_0}$. Which means that "most" subsets of $\mathbb R^n$ are not Borel sets. So it is not hard to show (or to believe if anything) that most mappings are not Borel measurable.

Now identify the continuous functions with $\Sigma^0_1$ sets. That is the preimage of an open set is an open set, this implies also that $\Pi^0_1$ sets are preserved (preimage of closed is closed). Furthermore, it is quite simple to show that if a function is continuous then the entire hierarchy is preserved. In particular, we only need to see what is the preimage of open sets, and from there the hierarchy will be preserved.

Now take the pointwise limit of continuous functions. This may not be continuous, but an open set will be at most $\Sigma^0_2$ or $\Pi^0_2$. We can iterate this, each time taking pointwise limits of anything previously defined. Since a pointwise limit of measurable functions is measurable we have that the union of $\aleph_1$ many steps of this construction yields only measurable functions.

On the other hand we can measure how high the basis for the topology is being transferred into the hierarchy of $\mathcal B_1$, and we can prove that applying this many pointwise limits iterations will give you this function.