# Category:Integral of Positive Measurable Function

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This category contains results about **Integral of Positive Measurable Function**.

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

## Subcategories

This category has the following 8 subcategories, out of 8 total.

### I

### T

- Tonelli's Theorem (4 P)

## Pages in category "Integral of Positive Measurable Function"

The following 14 pages are in this category, out of 14 total.

### I

- Integral of Characteristic Function/Corollary
- Integral of Increasing Function Composed with Measurable Function
- Integral of Increasing Function Composed with Measurable Function/Corollary
- Integral of Positive Measurable Function as Limit of Integrals of Positive Simple Functions
- Integral of Positive Measurable Function Extends Integral of Positive Simple Function
- Integral of Positive Measurable Function is Additive
- Integral of Positive Measurable Function is Monotone
- Integral of Positive Measurable Function is Positive Homogeneous
- Integral of Positive Measurable Function with respect to Restricted Measure
- Integral of Series of Positive Measurable Functions
- Integral of Survival Function
- Integral with respect to Discrete Measure
- Integral with respect to Series of Measures