Example of decreasing sequence of non-negative measurable functions such that $\int f_n \ d\mu= +\infty \ \text{ and } \ \int f \ d\mu=0.$

Question

Give an example of a decreasing sequence of non-negative measurable functions which are defined on the interval $[0,1]$ and which converge pointwise to a bounded function $f$ such that $$\int f_n \ d\mu= +\infty \ \text{ and } \ \int f \ d\mu=0.$$

Answer 1

$f_n(x)=\frac{1}{nx^2}, f(x)=0$ then $$\int f_n(x)dx=\frac{1}{n}\int_0^1\frac{1}{x^2}dx=\frac{1}{n}\left[-\frac{1}{x}\right]_0^1 =\infty \quad and \quad \int f(x) dx=\int0dx=0$$