Let $\displaystyle A\subseteq[0,1], |A|=\aleph_0$,

f(x)={0 if $\displaystyle x\notin A$, 1 if $\displaystyle x\in A$}

Prove that f is not integrable in [0,1], or show a negative proof and prove that it is integrable in [0,1].

Printable View

- May 5th 2010, 01:33 AMadam63Integrability
Let $\displaystyle A\subseteq[0,1], |A|=\aleph_0$,

f(x)={0 if $\displaystyle x\notin A$, 1 if $\displaystyle x\in A$}

Prove that f is not integrable in [0,1], or show a negative proof and prove that it is integrable in [0,1]. - May 5th 2010, 02:05 AMHallsofIvy
- May 5th 2010, 02:25 AMadam63
- May 6th 2010, 02:25 AMadam63
- May 7th 2010, 09:41 PMadam63
Can anyone please help me with that? I need it urgently.

- May 7th 2010, 09:52 PMDrexel28
- May 7th 2010, 09:54 PMDrexel28
Just because your perturbing the zero function at a countable number of places does not mean the set of all discontinuities created is countable, and thus necessarily of measure zero. The indicator function for the rationals differs from the zero function at countably many places but the set of discontinuities introduced has measure one.