May I plz get some assistance on the following question;

If f and g are integrable on a closed set and f(x)<=h(x)<=g(x) for all x in in that set then h (a function) is integrable on that set.

Printable View

- May 12th 2009, 12:31 AMnankorProve or disprove integration proof
May I plz get some assistance on the following question;

If f and g are integrable on a closed set and f(x)<=h(x)<=g(x) for all x in in that set then h (a function) is integrable on that set. - May 12th 2009, 03:09 AMPlato
- May 12th 2009, 05:03 AMputnam120
The statement is false.

Take $\displaystyle f(x)=0$ and $\displaystyle g(x)=1$ on the interval $\displaystyle [0,1]$ and $\displaystyle h(x)=1$ if $\displaystyle x\in\mathbb{Q}$ and 0 other wise. This shows that the statement is false for the Riemann integral.

For Lebesgue integrable take $\displaystyle h(x)=1$ on a set $\displaystyle \mu\subset[0,1]$ and $\displaystyle \mu$ is not Lebesgue measurable, and 0 else where. Then $\displaystyle h(x)$ is not Lebesgue integrable.