with Lebesgue measure, or more generally any Borel measure, then all continuous functions are measurable. In fact, practically any function that can be described is measurable.

Are continuous functions Borel measurable?

In particular, every continuous function between topological spaces that are equipped with their Borel σ-algebras is measurable.

How do you know if a function is measurable?

To prove that a real-valued function is measurable, one need only show that {ω : f(ω) < a}∈F for all a ∈ D. Similarly, we can replace < a by > a or ≤ a or ≥ a. Exercise 10. Show that a monotone increasing function is measurable.