Show that there exists measurable functions f_n defined on some measure subspace, st f_n-> f a.e. but such that f is not measurable.

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- November 6th 2008, 07:32 PMSmilingnon-measurable function
Show that there exists measurable functions f_n defined on some measure subspace, st f_n-> f a.e. but such that f is not measurable.

- November 7th 2008, 12:56 AMOpalg
- November 7th 2008, 01:10 AMSmiling
I think that the measure here has to be a non Lebesgue measure. I was thinking about a Borel measure in the Cantor set. We know that there is a Borel non-measurable subset of the Cantor set. But, I don't know what function to pick....

- November 7th 2008, 02:50 AMOpalg
In that case, what do you mean by a.e. convergence? If it means almost everywhere with respect to the Borel measure, then surely the usual proof that a (pointwise a.e.) limit of a sequence of measurable functions is measurable will work for that measure? On the other hand, if it means a.e. (Lebesgue) convergence then I think you can have a rather trivial example, like this. Let f be the characteristic function of a Borel non-measurable subset B of the Cantor set, and let f_n be the zero function, for all n. Then f_n → f a.e. (because f is also zero a.e.!), each f_n is Borel measurable, but f is not.