Monotone Convergence theorem

Question

Give an example of a sequence of Lebesgue integrable functions $\{f_{n}\}$ converging everywhere to a Lebesgue integrable function $f$ such that

$$\ \lim_{n \to \infty} \int_{-\infty}^{+\infty} f_{n} (x) dx <\int_{-\infty}^{+\infty} f (x) dx $$

Answer 1

There are different counterexamples, two of which being:

• the mass goes to infinity: $$f_n(x) = \begin{cases} -1 &\text{if} & n<x<n+1\\ 0 &\text{otherwise} \end{cases}$$

• the mass concentrates around one point: $$f_n(x) = \begin{cases} -n &\text{if} & 0<x<\frac 1n\\ 0 &\text{otherwise} \end{cases}$$