I am not sure if I understand the $\inf$ meanings in Fatou's Lemma. The assertion states that:
$$\lim_{n \to \infty} \int_X \inf f_n d\mu \leq \lim \inf_{n \to \infty} \int_{X} f_n d\mu$$
I would like to understand its meaning and see the use of the lemma through a simple example. So let the sequence of functions be $f_n(x) = \big(1 + \frac{1}{n} \big)^n x$ for $n \geq 1$, where $n \in \mathbb{N}$.
Obviously it is an increasing sequence that converges to $f(x) = e x$. Let us integrate over the $[0,1]$ interval. My questions are:
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For the left hand side of the inequality we would need to integrate $\inf f_n$ and then evaluate its limit. This function would in our case be given by $x$ right?. If true, then the value of the integal is simply $\frac{1}{2}$. What do we do with the limit then?
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And for the right hand side we have to evaluate $\lim \inf_{n \to \infty} \frac{1}{2} \big(1 + \frac{1}{n} \big)^n$. The smaller value $\forall n \geq 1$ is $1$. Again, what is the use of the limit here?
Thanks!
Best Answer
As people have pointed out in the comments, you are confusing $\liminf$ and the infimum notation. For a concrete example of strict inequality, consider $f_n = n\chi_{[0,1/n]}$ on $[0,1]$.