Thread: 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?

Originally Posted by tianye

{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 (or ) is measurable, for any sequence of measurable functions. It follows that if the sequence has a pointwise limit then that must be measurable.

Originally Posted by Opalg

Yes. The usual way to prove this is to show that (or ) 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?

Originally Posted by tianye

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.

Originally Posted by tianye

The family of functions is indexed by an uncountable set.Does the sequence converge to a Lebesgue measurable function in this condition?

In particular, is also the pointwise limit of the sequence , so that it is measurable.

Originally Posted by Laurent

In particular, is also the pointwise limit of the sequence , so that it is measurable.

It's easy when you look at it that way!