Difference between revisions of "Integral of a positive function (measure theory)"

Latest revision as of 19:18, 14 April 2017

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.

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

• $\int f(x)\mu(\mathrm{d}x)$^[1]
• $\int f(x)\mathrm{d}\mu(x)$^[1]

Immediate results

Notes

1. Jump up ↑ So $f:X\rightarrow\bar{\mathbb{R} }^+$
2. 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)
3. Jump up ↑ The $g\le f$ 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

References

1. ↑ ^{Jump up to: 1.0 1.1 1.2 1.3} Measures, Integrals and Martingales - René L. Schilling