let {$f_n$} be sequence of integrable functions defined on an interval $[a,b]$. Then
a) If $f_n(x)\to 0$ a.e., then $\int_{a}^{b}f_n(x)dx \to 0$
b) If $\int_{a}^{b}f_n(x)dx \to 0$ then $f_n(x)\to 0$ a.e.
c) if $f_n(x)\to 0$ a.e.and each $f_n$ is bounded function, then $\int_{a}^{b}f_n(x)dx \to 0$
d) if $f_n(x)\to 0$ a.e and $f_n$'s are uniformly bounded then $\int_{a}^{b}f_n(x)dx \to 0$
I have no idea how to look for the solution of above?
Statement (a) is false: let $[a,b]=[0,1]$, and define $f_n(x)=n\chi_{[0,\frac{1}{n}]}$.
Statement (b) is false, even if the $f_n$ are required to be non-negative. Again let $[a,b]=[0,1]$, and define a sequence by $f_1=\chi_{[0,\frac{1}{2}]}$, $f_2=\chi_{[\frac{1}{2},1]}$, $f_3=\chi_{[0,\frac{1}{4}]}$, $f_4=\chi_{[\frac{1}{4},\frac{1}{2}]}$, and so on. This is sometimes called the "typewriter sequence" or the "floating, shrinking interval".
Statement (c) is false. The same example as for statement (a) works.
Statement (d) is true for Lebesgue measure on a compact interval $[a,b]$, by the Dominated Convergence Theorem.