Which of the following imply $\lim_{n \to \infty} \int_{0}^1 f_n(x)dx= \int_{0}^1 f(x)dx$? let $\{f_n\}$ be sequence of non-negative continuous functions defined on $[0,1]$. Assume that $f_n(x) \to f(x)$ for every $x \in [0,1]$. which of the following condition imply that $$\lim_{n \to \infty} \int_{0}^1 f_n(x)dx= \int_{0}^1 f(x)dx$$
a. $f_n(x) \uparrow f(x)$ for every $x \in [0,1]$.
b. $f_n(x) \le f(x)$ for every $x \in [0,1]$
c. $f$ is continuous.  

as i had tried i found a counter example for option(c)and it is $$f(x)=\begin{cases} n^{2}x &\text {if}\:  0 \le x \le 1/n\\ -n^{2}(x-2/n) &\text{if}\;1/n \le x \le 2/n\\ 0 &\text{if}\;  2/n \le x \le 1\end{cases}$$so (c) is not true.
  for option (a) i used the theorem Monotone  convergence theorem so (a) is true but dont know how deal with option (c).


If $f_n(x) \le f(x)$ for each $x$, then $\limsup_n \int f_n(x)dx \le \int f(x)dx$. Since $f_n \to f$, Fatou implies $\liminf_n \int f_n(x)dx \ge \int f(x)dx$. So (b) works.
