Where and Why do we use measurable functions in modern probability theory?

Question

I am a newbie to the measure theoretic framework for probability. While studying MMSE Estimation I came across the term "borel function" which upon further googling made me understand that it's a "measurable function". I understand the definition of a measurable function but what I fail to grasp is that how is this concept of "borel function" or for that matter "measurable function" important in the theory of Minimum Mean Square Error(MMSE) Estimation. Can anyone please elaborate on this?

Thanks in advance.

Answer 1

Actually, every function that you can write down explicitly, or are likely to encounter in real life, is a measurable function. In order to make the theory go smoothly, you need to assume that you're not dealing with pathological examples for which it is impossible to define probabilities.