- #1
happysauce
- 42
- 0
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[itex]_{[0,1]}[/itex]×P([0,1]) where B[itex]_{[0,1]}[/itex] 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[itex]\in[/itex]N and f[itex]^{-1}[/itex](E) [itex]\in[/itex] M, then f is (M,N) - measurable.
So for the first question all I know is that E[itex]\in[/itex]B[itex]_{ℝ}[/itex] and f[itex]^{-1}[/itex](E) [itex]\in[/itex] M, and F[itex]\in[/itex] B[itex]_{ℝ}[/itex] and g[itex]^{-1}[/itex](F) [itex]\in[/itex] N