Measure theory (Floris' talk)

Stream: Lean Together 2021

[Kevin Buzzard]

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]

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]

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.