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

I am a beginner in the theoretical framework of measurement for probability. While studying the MMSE estimate, I came across the term "borel function" which, after searching on Google, made me understand that it was a "measurable function". I understand the definition of a measurable function, but I do not know what makes this concept of "borel function" or, essentially, "measurable function" important in the theory of estimating Mean squared error (MMSE). Can any one please elaborate on this topic?

Thank you in advance.