Thread: non-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.

Originally Posted by Smiling

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.

Not true. A (pointwise a.e.) limit of a sequence of measurable functions is always measurable.

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....

Originally Posted by Smiling

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....

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.