# The general form of a measurable set in a product measurable space

Hi everyone: Suppose that $(X,\mathfrak{M},\mu)$ and $(Y,\mathfrak{N},\nu)$ are two measure spaces and consider the product measure space $(X\times Y,\sigma(\mathfrak{M}\times\mathfrak{N}),\mu\times\nu)$.

1. What is the general form of a measurable set in the product measure space?
2. Is it always in the form $E\times F$ with $E$ in $\mathfrak{M}$ and $F\in\mathfrak{N}$?
3. Is it contained in such product of measurable sets?

4. Is it a union of such product of measurable sets?

5. Does it contain such product of measurable sets? What exactly? Thanks for your help.

• No. Yes, take $X\times Y$. No. Yes, take $\varnothing\times\varnothing$. – Stefan Hansen Aug 10 '14 at 19:22

1. It is hard to give a complete description of the product $\sigma$-algebra. It will certainly contain any countable union of products of measurable sets, but the collection of such sets won't be stable by complementation.
2. Not necessarily, for example if $X=Y=\{0,1\}$ where all subsets are measurable, then $\{(0,1),(1,0)\}$ is an element of the product $\sigma$-algebra which is not a product of two subsets.
3. In general, we can't say anything non-trivial: $X\times X$. For example, $X=Y=\mathbf Z$ endowed with the power set and $S:=\{(x,x),x\in\mathbf Z\}$.
4. If you mean a countable union (otherwise it's trivial), not necessarily. Take $X=Y=[0,1]$ with Borel $\sigma$-algebra and $S=\{(x,x),x\in [0,1]\}$.