{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.
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?