Let $h_n$ be a sequence of measurable functions. I managed to prove the following equalities (measurability is not relevant in this case):
$$\left\{{\inf_{n \in \mathbb N}}{h_n} \geq c \right\} = \bigcap_{n=1}^{\infty} \left\{ h_n \geq c \right\},$$
$$\left\{{\sup_{n \in \mathbb N}}{h_n} \leq c \right\} = \bigcap_{n=1}^{\infty} \left\{ h_n \leq c \right\}$$
and therefore the equalities of the respective complements:
$$\left\{{\inf_{n \in \mathbb N}}{h_n} < c \right\} = \bigcup_{n=1}^{\infty} \left\{ h_n < c \right\},$$
$$\left\{{\sup_{n \in \mathbb N}}{h_n} > c \right\} = \bigcup_{n=1}^{\infty} \left\{ h_n > c \right\}$$
However, I cannot prove that, for example,
$$\left\{{\inf_{n \in \mathbb N}}{h_n} > c \right\} = \bigcap_{n=1}^{\infty} \left\{ h_n > c \right\},$$ or $$\left\{{\inf_{n \in \mathbb N}}{h_n} \leq c \right\} = \bigcup_{n=1}^{\infty} \left\{ h_n \leq c \right\} $$
The last equalities first came up in my tutorial class and I also found them on page 20 (8) of this pdf: http://www.applebaum.staff.shef.ac.uk/Ch2MeasFn.pdf. Can someone clarify whether these equalities hold and, if so, give a detailed demonstration?
Consider the constant functions $h_n = \frac{1}{n}$ and $c=0$. Then we have \begin{align} \left\{ \inf_{n} h_n \le 0 \right\} = \left\{ \inf_{n} \frac{1}{n} \le 0 \right\} = \Omega. \end{align} On the other hand we have \begin{align} \bigcup_{n} \left\{ h_n \le 0 \right\} = \bigcup_{n} \left\{ \frac{1}{n} \le 0 \right\} = \emptyset. \end{align} So the equalities do not hold.