Real Analysis HELP: Measurable Functions on Measure Subspace

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Homework Statement

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.

Homework Equations

Converges a.e. means that converges everywhere except on a set of measure zero.

The Attempt at a Solution

Need to construct a measure space in which some subset of a measurable set of measure zero is not measurable. However, such measure space is not compelte.

 

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I know that I need to construct a measure space in which some subset of a measurable set of measure zero is not measurable. However, such measure space is not complete.
It seems that there is a subset of the Cantor set that is not borel measurable...so, if you choose the Borel measure, then you know it is not complete and that m(Cantor set)=0...
I am not sure how to choose the function though...maybe choose the Cantor function?