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## Homework Statement

Let {f

_{n}} be a sequence of measurable functions defined on a measurable set E. Define E

_{0}to be the set of points x in E at which {f

_{n}(x)} converges. Is the set E

_{0}measurable?

## Homework Equations

Proposition 2:

Let the function f be defined on a measurable set E. Then, f is measurable if and only if for each open set O, the inverse image of O under f, f

^{-1}(O) = {x[itex]\in[/itex]E | f(x) [itex]\in[/itex] 0}, is measurable.

## The Attempt at a Solution

Since E

_{0}= {x[itex]\in[/itex]E| {f

_{n}(x)} converge}, then {f

_{n}(x)} [itex]\in[/itex] O and by Proposition 2, E

^{-1}

_{0}is measurable. But, that doesn't mean that E is measurable...

Isn't it true that continuous E

^{-1}being measurable implies E is measurable? Is that where I should go with this?

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