Read the following proposition in a source. Can anyone supply a proof (or general direction)?
"If a non-negative function $f$ is measurable. There exists a sequence of increasing measurable step (or simple) functions, $\{s_n\}$, that converge to $f$ point-wise."
The book then states that, with an additional condition, "$\int{s}d\mu$ is finite for every step function $s\in\{s_n\}$", we have $\int f d\mu=sup\{\int sd\mu\}$.
I suppose the existence of the supremum is due to Least Upper Bound theorem? And the condition "$\int{s}d\mu$ is finite for every step function $s\in\{s_n\}$" is actually saying the sequence of Lebesgue integral of step functions is bounded above? I'm uncertain because the language is vague: I can say $a_n$ is finite for every n in sequence {$a_n$} while the sequence can still be unbounded right?