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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.

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    $\begingroup$ No. Yes, take $X\times Y$. No. Yes, take $\varnothing\times\varnothing$. $\endgroup$ – Stefan Hansen Aug 10 '14 at 19:22
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  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 $S=\{(x,x),x\in [0,1]\}$.

However, if we have finite measure space, a measurable set for the product measure can be approximated in the sense of the product measure by finite unions of products of measurable sets.

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  • $\begingroup$ Could you please give a reference for the last assertion? $\endgroup$ – folouer of kaklas Feb 3 '17 at 16:34

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