The integral of a nonnegative measurable function definition: what does $f_n(\omega)\uparrow f(\omega)$ mean? Let $f:\Omega\to\overline{\mathbb{R}}_{+}$ be a nonnegative measurable function on $(\Omega,\mathcal{F},\mu)$. The integral of $f$ with respect to $\mu$ is defined as
\begin{align*}
\int f \, \mathrm{d}\mu = \lim_{n\to\infty}\int f_n \, \mathrm{d}\mu
\end{align*}
where $\{f_n\}_{n\geq1}$ is any sequence of nonnegative simple functions s.t. $f_n(\omega)\uparrow f(\omega)$ for all $\omega\in\Omega$.
MY QUESTION
What is the meaning of the notation $f_n(\omega)\uparrow f(\omega)$?
Does this mean that $f_{n}$ converges pointwise to $f$ and $f_{n+1}(\omega)\geq f_n(\omega)$ for every $\omega\in\Omega$?
I am new to this so any help is appreciated.

Yes, it's as you say. This means that the sequence of functions converges "upwards" to $f$, meaning that $f_n\to f$ pointwise and for each $x\in \Omega$, and for each $n$, $f_n(x)\le f_{n+1}(x)$.
