Integral of a positive function (measure theory)/Definition

Definition

Let $(X,\mathcal{A},\mu)$ be a measure space, the $\mu$ -integral of a positive numerical function, $f\in\mathcal{M}^+_{\bar{\mathbb{R} } }(\mathcal{A})$ ^{[Note 1]}^{[Note 2]} is^[1]:

$\int f\mathrm{d}\mu:=\text{Sup}\left\{I_\mu(g)\ \Big\vert\ g\le f, g\in\mathcal{E}^+(\mathcal{A})\right\}$ ^{[Note 3]}

Recall that:

$I_\mu(g)$ denotes the $\mu$ -integral of a simple function
$\mathcal{E}^+(\mathcal{A})$ denotes all the positive simple functions in their standard representations from $X$ considered with the $\mathcal{A}$ $\sigma$ -algebra.

TODO: Link to $\mathcal{E}$ somewhere, are they numeric or real valued?

TODO: Can every simple function be made into a standard representation, thus what is $\mathcal{E}$ exactly and what is the domain of $I_\mu$ exactly?

There are alternate notations, that make the variable of integration more clear, they are:

Jump up ↑ So $f:X\rightarrow\bar{\mathbb{R} }^+$
Jump up ↑ Notice that $f$ is $\mathcal{A}/\bar{\mathcal{B} }$ -measurable by definition, as $\mathcal{M}_\mathcal{Z}(\mathcal{A})$ denotes all the measurable functions that are $\mathcal{A}/\mathcal{Z}$ -measurable, we just use the $+$ as a slight abuse of notation to denote all the positive ones (with respect to the standard order on $\bar{\mathbb{R} }$ - the extended reals)
Jump up ↑
The g≤f is an abuse of notation for saying that g is everywhere less than f, we could have written:
- $\int f\mathrm{d}\mu=\text{Sup}\left\{I_\mu(g)\ \Big\vert\ g\le f, g\in\mathcal{E}^+\right\}=\text{Sup}\left\{I_\mu(g)\ \Big\vert\ g\in\left\{h\in\mathcal{E}^+(\mathcal{A})\ \big\vert\ \forall x\in X\left(h(x)\le f(x)\right)\right\}\right\}$ instead.
Inline with: Notation for dealing with (extended) real-valued measurable maps