Sequence of Measurable Functions

[jdcasey9]

Homework Statement

Let {f_n} 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 {f_n(x)} converges. Is the set E₀ 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$\in$E | f(x) $\in$ 0}, is measurable.

The Attempt at a Solution

Since E₀ = {x$\in$E| {f_n(x)} converge}, then {f_n(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?

 

[Dick]

Homework Statement

Let {f_n} 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 {f_n(x)} converges. Is the set E₀ 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$\in$E | f(x) $\in$ 0}, is measurable.

The Attempt at a Solution

Since E₀ = {x$\in$E| {f_n(x)} converge}, then {f_n(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?

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.