Proving functions in product space are measurable.

[happysauce]

Homework Statement

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])

Homework Equations

Don't know any.

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\inN and f^{-1}(E) \in M, then f is (M,N) - measurable.

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

 

[haruspex]

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

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