Integral (measure theory)

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Definition

Given a measure space, $(X,\mathcal{A},\mu)$ and a function $f:X\rightarrow\bar{\mathbb{R} }$, we say that $f$ is $\mu$-integrable if:

• $f$ is a measurable map, an $\mathcal{A}/\bar{\mathcal{B} }$-measurable map; and if
• The integrals $\int f^+\mathrm{d}\mu,\ \int f^-\mathrm{d}\mu<\infty$, then:

We define the $\mu$-integral of $f$ to be:

• $$\int f\mathrm{d}u:=\int f^+\mathrm{d}\mu-\int f^-\mathrm{d}\mu$$

Where:

• $$\int f^\pm\mathrm{d}\mu$$ are the integrals of positive functions.

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Reminder: Integration of positive functions

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

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

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.

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Reminder: Integration of simple functions

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For a simple function in its standard representation, say $f:=\sum^n_{i=0}x_i\mathbf{1}_{A_i}$ then the $\mu$-integral, $I_\mu:\mathcal{E}^+\rightarrow\mathbb{R}$ is:

• $$I_\mu(f):=\sum^n_{i=1}x_i\mu(A_i)\in[0,\infty]$$

Note that this is independent of the particular standard representation of $f$.

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Reminder: Definition of simple function

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A simple function $f:X\rightarrow\mathbb{R}$ on a measurable space $(X,\mathcal{A})$ is a:

• function of the form $\sum^N_{i=1}x_i\mathbf{1}_{A_i}(x)$ for
• finitely many sets, $A_1,\ldots,A_N\in\mathcal{A}$ and
• finitely many $x_1,\ldots,x_n\in\mathbb{R}$