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 fd\mu = \lim_{n\to\infty}\int f_{n}d\mu \end{align*}
where $\{f_{n}\}_{n\geq 1}$ is any sequence of nonnegative simple functions such that $f_{n}(\omega)\uparrow f(\omega)$ for all $\omega$.
MY QUESTION
What is the importance of the measure space $(\Omega,\mathcal{F},\mu)$ and the measurability of $f$ in the proposed definition?
As far as I have understood, we need a domain, which is given by $\Omega$; we need a $\sigma$-algebra $\mathcal{F}$ in order to properly define the simple functions $f_{n}$; and we need the measure $\mu$ in order to integrate each $f_{n}$ so that we can take the limit.
Am I missing something? Please let me know if it is the case.