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Math Help - If f necessarily Lebesgue measurable?

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    If f necessarily Lebesgue measurable?

    {ft},t varies from 0 to infinity, is a family of Lebesgue measurable functions that converges pointwise to f. Is f necessarily Lebesgue measurable?
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    Quote Originally Posted by tianye View Post
    {ft},t varies from 0 to infinity, is a family of Lebesgue measurable functions that converges pointwise to f. Is f necessarily Lebesgue measurable?
    Yes. The usual way to prove this is to show that \limsup_{n\to\infty}f_n (or \liminf_{n\to\infty}f_n) is measurable, for any sequence of measurable functions. It follows that if the sequence has a pointwise limit then that must be measurable.
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    Have you noticed that?

    Quote Originally Posted by Opalg View Post
    Yes. The usual way to prove this is to show that \limsup_{n\to\infty}f_n (or \liminf_{n\to\infty}f_n) is measurable, for any sequence of measurable functions. It follows that if the sequence has a pointwise limit then that must be measurable.
    The family of functions is indexed by an uncountable set.Does the sequence converge to a Lebesgue measurable function in this condition?
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    Quote Originally Posted by tianye View Post
    The family of functions is indexed by an uncountable set.Does the sequence converge to a Lebesgue measurable function in this condition?
    I didn't realise that the index set was supposed to be the reals. In that case, I don't know whether the limit function has to be measurable.
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    Quote Originally Posted by tianye View Post
    The family of functions is indexed by an uncountable set.Does the sequence converge to a Lebesgue measurable function in this condition?
    In particular, f is also the pointwise limit of the sequence (f_n)_{n\in\mathbb{N}}, so that it is measurable.
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    Quote Originally Posted by Laurent View Post
    In particular, f is also the pointwise limit of the sequence (f_n)_{n\in\mathbb{N}}, so that it is measurable.
    It's easy when you look at it that way!
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