I need a function for the following equality

Question

I need an example that there exists a measurable non-negative function $f_n:X\to\mathbb{R}$ which uniform converges to $f:X\to\mathbb{R}$, and $\displaystyle\lim_{n\to\infty} \int_X f_nd\mu$ exists, but $$\int_X fd\mu \neq \lim_{n\to\infty} \int_X f_n d\mu.$$

Answer 1

One classical example of what you want is a "delta-sequence". Let $f_n (x)= 2n \cdot \chi_{[-\frac 1 n,\frac 1 n]} (x), \space x \neq 0$ and $f_n(0)=0$. Note that $\int f_n \mathbb d x = 1$, therefore $\lim \limits _n \int f_n \mathbb d x = 1$. On the other hand, $\lim \limits _n f_n = 0$, therefore $\int \lim \limits _n f_n \mathbb d x = 0$.