# Thread: If f necessarily Lebesgue measurable?

1. ## 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?

2. 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 $\displaystyle \limsup_{n\to\infty}f_n$ (or $\displaystyle \liminf_{n\to\infty}f_n$) is measurable, for any sequence of measurable functions. It follows that if the sequence has a pointwise limit then that must be measurable.

3. ## Have you noticed that?

Originally Posted by Opalg
Yes. The usual way to prove this is to show that $\displaystyle \limsup_{n\to\infty}f_n$ (or $\displaystyle \liminf_{n\to\infty}f_n$) 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?

4. 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.

5. 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, $\displaystyle f$ is also the pointwise limit of the sequence $\displaystyle (f_n)_{n\in\mathbb{N}}$, so that it is measurable.

6. Originally Posted by Laurent
In particular, $\displaystyle f$ is also the pointwise limit of the sequence $\displaystyle (f_n)_{n\in\mathbb{N}}$, so that it is measurable.
It's easy when you look at it that way!