I was given the following homework problem.

Let $f:X\rightarrow \bar{\mathbb{R}}$ a set function and $X$ be a measurable space whose $\sigma$-algebra is generated by a finite partition $E_1,\dots ,E_n$. Describe all such measurable set functions.

Since unions and intersections commute, it seems measurable sets are just finite unions of $E_i$'s, but I don't see what to do with this - I can't extract any information about the fibers..

The previous question asked to describe measurable real functions out of an uncountable space whose $\sigma$-algebra is comprised of sets which are countable or cocountable. I found that measurability is equivalent to being constant on a cocountable set. I don't think this is related though.

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    $\begingroup$ Try to show that the measurable functions are precisely those that are constant on each $E_i $. $\endgroup$ – PhoemueX Feb 9 '16 at 15:16
  • $\begingroup$ @PhoemueX solution: Since the elements of the sigma algebra are just unions of $E_i$'s, each fiber must be such a union, and hence the function must be constant on $E_i$'s? That's all there is to it? $\endgroup$ – user311996 Feb 9 '16 at 15:23
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    $\begingroup$ Now, you have to show the reverse inclusion. There, you will need that the partition is finite (countable would be enough). $\endgroup$ – PhoemueX Feb 9 '16 at 15:39
  • $\begingroup$ @PhoemueX sorry, what reverse inclusion? Isn't it enough to show each fiber is a union of $E_i$'s? This is the same as saying it's measurable - where does cardinality come into play? $\endgroup$ – user311996 Feb 9 '16 at 16:40
  • $\begingroup$ You only know that a countable union of $E_i $s is measurable. If the partition $(E _i)_{i \in I} $ was uncountable (ie if $I $ was uncountable), it could be that a function which is constant on each $E_i $ is not measurable after all. $\endgroup$ – PhoemueX Feb 9 '16 at 17:11

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