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 $

  • 1
    $\begingroup$ 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. $\endgroup$ – Diego Fonseca Nov 19 '17 at 16:57

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 $$\int \limsup f_n \geqslant - \liminf (- \int f_n) = \limsup \int f_n $$


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.