Doubt in a step in the proof of Fatou's lemma

Question

I am reading a proof of Fatou's lemma and I don't follow a step. Consider a sequence of non-negative measurable functions on $X$ to $\overline{\mathbb{R}}$. Fatou's lemma states that

$$\int(\lim\inf f_n)\,d\mu\leq\lim\inf\int{}f_n\,d\mu$$

where $\mu$ is a measure on $X$. The proof begins by considering the sequence

$$g_m=\inf\{f_m,f_{m+1},f_{m+2},\ldots\}$$

therefore, if $m\leq{}n$ we have that

$$\int{}g_m\,d\mu\leq\int{}f_n\,d\mu$$

as long as $m\leq{}n$. The next step states that implies

$$\int{}g_m\,d\mu\leq\lim\inf\int{}f_n\,d\mu$$

I don't see this. How does this follow?

Answer 1

Informally, because you might chop off the minimum when you pass to a subset. That is, we have used the fact $$ F \subset G \implies \inf F \geq \inf G \text{.} $$

The expression $\liminf \int f_n \,\mathrm{d}\mu$ asks one to compute "the lower bound of the sequence of successive tails of the sequence". At some point, that sequence of successive tails is eventually (and then forever after) a subset of $F_m = \{\int f_m, \int f_{m+1}, \dots \}$. So we subsequently only consider subsets of $F_m$, and their infima can be no less than the infimum of the full set. I.e., $$ \inf F_m \leq \inf F_{m+1} \leq \inf F_{m+2} \leq \cdots $$