To express a integrable function as difference.

Question

Assume $f$ is an integrable function on $[0,1]$. I want to find functions $g$ and $h$, so that $f=g-h$ almost everywhere. The functions $g$ and $h$ should be pointwise limits of continuous functions $g_n$ and $h_n$, and both two sequences $g_n(x)$, $ h_n(x)$ are required to be increasing when $x$ is fixed.

Any help would be grateful, thanks.

@TooOldForMath I don't see why every integrable function should be Baire class 1. Example 1 in the first answer here is Riemann integrable but not of Baire class 1. — David Mitra, Feb 18 '14 at 11:26

score 1 · Answer 1 · edited Apr 13 '17 at 12:19

1

If such $g_n$, $h_n$ exist, then $f$ is the pointwise limit of the sequence $g_n-h_n$. Therefore, $f$ belongs to Baire class 1. David Mitra gave a link to an example of a Riemann integrable function that is not of Baire class 1.

Therefore, such $g_n$ and $h_n$ cannot be found in general.

edited Apr 13 '17 at 12:19

Community

1

answered Feb 19 '14 at 02:02

user127096

9,683

Oh, sorry for my mistake. I just want f=g-h almost everywhere. Will the proposition be correct this time? – lee Feb 20 '14 at 10:10

To express a integrable function as difference.

1 Answers1