# Fatou's Lemma in nonpositive function

Fatou's Lemma:

If $\{f_n\}$ is a sequence of non-negative measurable function, then $\int(\text{lim inf} \ \ f_n)\leq \text{lim inf}\int f_n$.

If $\{f_n\}$ is a sequence of non-positive measurable sequence of function, should the Fatou's Lemma be:

$$\int(\text{lim sup} \ \ f_n)\geq \text{lim sup}\int f_n$$

Is it correct?

I refer to the discusstion of this link Understanding proof of dominated convergence from Folland.

$\text{lim inf} \ \ -c_n = -\text{lim sup} \ \ c_n$

• We have to be careful, what does it mean to be non-positive? for me it is that not all of its values are positive, but it can take positive values, that is, not positive is not equal to negative. – Diego Fonseca Nov 19 '17 at 16:57

## 1 Answer

This is correct. Using hat you said about the limsups,
$$\int \limsup f_n = \int - \liminf (-f_n)$$ As $-f_n$ is positive, so applying Fatou one gets $\int \liminf (-f_n) \leqslant \liminf (-\int f_n)$, so $$\int \limsup f_n \geqslant - \liminf (- \int f_n) = \limsup \int f_n$$