# Zulip Chat Archive

## Stream: Lean Together 2021

### Topic: Measure theory (Floris' talk)

#### Kevin Buzzard (Jan 04 2021 at 15:48):

I was totally confused about a part of your talk Floris -- how could a totally random non-continuous non-measurable function from a measure space X to the norm 1 elements of a Banach space E be approximated by anything? (its norm is fine, but the function itself is pathological). That was my question. Jason Rute just suggested that I was supposed to be restricting to measurable functions though.

#### Floris van Doorn (Jan 04 2021 at 16:03):

I should have been more clear: the L1-functions indeed consist of functions that are measurable + integrable.

I believe we use (roughly?) the following construction to approximate any measurable + integrable function `f`

:

- Enumerate a dense set on your Banach space E
`d : nat -> E`

. - The
`n`

-th approximation of your function sends`x`

to the point among`{d 0, ... d n}`

that is closest to`f x`

. - Because
`f`

is integrable, this will converge in the L1-norm to`f`

as`n -> \infty`

#### Floris van Doorn (Jan 04 2021 at 16:04):

I *think* this even works for functions `f`

that are non-measurable (but have a finite integral). The thing you don't get in this case is that the simple functions have measurable preimages.

Last updated: Dec 20 2023 at 11:08 UTC