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.
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.