# Preimage of a Lebesgue (non-Borel) measurable set

This is an textbook exercise: If $$(\Omega, \mathcal{F}, \mu)$$ is a complete metric space, then a function $$f:\Omega \rightarrow \mathbb{R}$$ is measurable iff sets of the form {$$\omega: | f(\omega)\le t, t\in \mathbb{R}$$} belong to $$\mathcal{F}$$. Here $$\mathbb{R}$$ is equipped with the sigma algebra of Lebesgue measurable sets, $$\mathcal{M(\mathbb{R})}$$.

My attempt: I want to show that $$\forall M\in \mathcal{M(\mathbb{R})}$$, $$f^{-1}(M)\in \mathcal{F}$$. I know that $$M$$ is the union of a Borel set $$B$$ and a null set $$N$$ in $$\mathbb{R}$$. So I just need to show $$f^{-1}(N)\in \mathcal{F}$$ for any null set $$N$$. But I don't know how to use the completeness of $$\mathcal{F}$$. If I can show there exists a Borel set $$\tilde{B}$$ such that $$\tilde{B} \supset N$$ meanwhile $$f^{-1}(\tilde{B})$$ is null in $$(\Omega, \mathcal{F}, \mu)$$ then it's done, but I'm not sure if this is possible.

I know that Lebesgue measurable function is almost a Borel measurable function but both definitions use preimage of Borel sets only, I'm not sure how to deal with preimage of a Lebesgue measurable set.

I have a feeling this should be pretty straightforward but I wasn't able to make progress. Any hint is appreciated.

So take a continuous function $$f:(\mathbb{R},\mathcal{M}(\mathbb{R}),\lambda) \rightarrow (\mathbb{R},\mathcal{M}(\mathbb{R}),\lambda)$$. The domain is a complete measure space, and since intervals are Borel the inverse image of intervals are Borel and hence in $$\mathcal{M}(\mathbb{R})$$, but the preimage of a general Lebesgue measurable set may not be in $$\mathcal{M}(\mathbb{R})$$.