Sequence of Measurable Functions

1. Oct 20, 2012

jdcasey9

1. The problem statement, all variables and given/known data

Let {fn} be a sequence of measurable functions defined on a measurable set E. Define E0 to be the set of points x in E at which {fn(x)} converges. Is the set E0 measurable?

2. Relevant 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$\in$E | f(x) $\in$ 0}, is measurable.

3. The attempt at a solution

Since E0 = {x$\in$E| {fn(x)} converge}, then {fn(x)} $\in$ O and by Proposition 2, E-10 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?

Last edited: Oct 20, 2012
2. Oct 20, 2012

Dick

Have you shown lim inf and lim sup of a sequence of measurable functions are measurable (possibly with values in the extended reals)? The set of convergent points are where those two functions are equal.