# Proving functions in product space are measurable.

1. Dec 2, 2012

### happysauce

1. The problem statement, all variables and given/known data
I have a lot of questions that ask me to prove certain functions are measureable.

For example I have to show that given f:X→ ℝ is M - measurable and g:Y→ ℝ is N - measurable
implies that fg is M×N measurable.

Another is prove that f = {1 when x=y, 0 else} is measurable on B$_{[0,1]}$×P([0,1]) where B$_{[0,1]}$ is the borel sets on [0,1] with respect to lebesgue measure and the measure P([0,1]) is the counting measure (cardinality of a set in [0,1])

2. Relevant equations
Don't know any.

3. The attempt at a solution

I don't have a clue what to do because I don't know any definition of measurable functions in a product space. I know the case for single measure spaces, If E$\in$N and f$^{-1}$(E) $\in$ M, then f is (M,N) - measurable.

So for the first question all I know is that E$\in$B$_{ℝ}$ and f$^{-1}$(E) $\in$ M, and F$\in$ B$_{ℝ}$ and g$^{-1}$(F) $\in$ N

2. Dec 2, 2012

### haruspex

Yes, but you are free to choose E=F here. What will the preimage of that be under (f, g)?