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Sequence of Measurable Functions

  1. Oct 20, 2012 #1
    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[itex]\in[/itex]E | f(x) [itex]\in[/itex] 0}, is measurable.

    3. 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?
     
    Last edited: Oct 20, 2012
  2. jcsd
  3. Oct 20, 2012 #2

    Dick

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    Homework Helper

    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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