Fatou's Lemma Proof Misunderstanding

I am missing something fairly basic in the proof for Fatou's lemma.

Statement

Suppose $<f_n>$ is a sequence of non-negative measurable functions, such that $f_n\to f$ almost everywhere. Then

$$\int f \leq \underline{\lim} \int f_n$$

Proof

Suppose as above. Further, let $h$ be a bounded measurable function which is not greater than $f$ and which vanishes outside a set $E'$ of finite measure. Define

$$h_n = \min\{h,f_n\}$$

for each $x$. Then

$$\int_E h = \int_{E'} h = \lim_{n \to \infty} \int_{E'} h_n \leq \underline{\lim} \int f_n$$

Taking the supremum over $h$ gives the result.

Question

I do not see how taking the supremum over $h$ gives the result. Do they mean $h$ as a function or $h$ over $x$?

• What is $f$ in your statement? – uniquesolution Jan 3 '18 at 17:00
• $lim_{n \to \infty} f_n$ – Aaron Zolotor Jan 3 '18 at 17:44
• Tell us what $f$ is in the statement – zhw. Jan 3 '18 at 18:22
• They mean over bounded measurable functions $h$. – jgon Jan 3 '18 at 20:00