Definition:Integral of Positive Measurable Function
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Definition
Let $\struct {X, \Sigma, \mu}$ be a measure space.
We define the $\mu$-integral of positive measurable functions, denoted $\ds \int \cdot \rd \mu: \MM_{\overline \R}^+ \to \overline \R_{\ge 0}$, as:
- $\forall f \in \MM_{\overline \R}^+: \ds \int f \rd \mu := \sup \set {\map {I_\mu} g: g \le f, g \in \EE^+}$
where:
- $\MM_{\overline \R}^+$ denotes the space of positive $\Sigma$-measurable functions
- $\overline \R_{\ge 0}$ denotes the positive extended real numbers
- $\sup$ is a supremum in the extended real ordering
- $\map {I_\mu} g$ denotes the $\mu$-integral of the positive simple function $g$
- $g \le f$ denotes pointwise inequality
- $\EE^+$ denotes the space of positive simple functions
Integral Sign
The symbol:
- $\ds \int \ldots \rd \mu$
is called the integral sign.
Note that there are two parts to this symbol, which embrace the function $f$ which is being integrated.
Also known as
Sometimes it is convenient to indicate the integration variable explicitly. In these cases, one may write one of:
- $\ds \int \map f x \map \mu {\d x}$
- $\ds \int \map f x \map {\d \mu} x$
in place of $\ds \int f \rd \mu$.
Sometimes it is also presentationally convenient to write $\map \mu f$.
It should be noted that this is abuse of notation, since a measure does not take functions as arguments.
Also see
- Integral of Characteristic Function: Corollary
- Integral of Positive Measurable Function is Positive Homogeneous
- Integral of Positive Measurable Function is Additive
- Integral of Positive Measurable Function is Monotone
Sources
- 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next) $9.4$