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Sequence of Measurable Functions

  • Thread starter jdcasey9
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Homework Statement



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?

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 E0 = {x[itex]\in[/itex]E| {fn(x)} converge}, then {fn(x)} [itex]\in[/itex] 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?
 
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Answers and Replies

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Dick
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Homework Statement



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?

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 E0 = {x[itex]\in[/itex]E| {fn(x)} converge}, then {fn(x)} [itex]\in[/itex] 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.
 

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