# Category:Measures

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This category contains results about measures.

Let $\struct {X, \Sigma}$ be a measurable space.

Let $\mu: \Sigma \to \overline \R$ be a mapping, where $\overline \R$ denotes the set of extended real numbers.

Then $\mu$ is called a **measure** on $\Sigma$ if and only if $\mu$ has the following properties:

\((1)\) | $:$ | \(\ds \forall E \in \Sigma:\) | \(\ds \map \mu E \) | \(\ds \ge \) | \(\ds 0 \) | |||

\((2)\) | $:$ | \(\ds \forall \sequence {S_n}_{n \mathop \in \N} \subseteq \Sigma: \forall i, j \in \N: S_i \cap S_j = \O:\) | \(\ds \map \mu {\bigcup_{n \mathop = 1}^\infty S_n} \) | \(\ds = \) | \(\ds \sum_{n \mathop = 1}^\infty \map \mu {S_n} \) | that is, $\mu$ is a countably additive function | ||

\((3)\) | $:$ | \(\ds \exists E \in \Sigma:\) | \(\ds \map \mu E \) | \(\ds \in \) | \(\ds \R \) | that is, there exists at least one $E \in \Sigma$ such that $\map \mu E$ is finite |

## Subcategories

This category has the following 18 subcategories, out of 18 total.

### A

### C

- Convolution of Measures (2 P)
- Counting Measure (3 P)

### D

- Diffuse Measures (2 P)
- Discrete Measure (1 P)

### F

### I

- Invariant Measures (1 P)

### K

### M

- Measure is Subadditive (2 P)
- Measure with Density (1 P)

### N

### P

- Probability Measures (5 P)
- Pushforward Measures (2 P)

### R

- Restricted Measures (2 P)

### S

- Series of Measures (2 P)
- Sigma-Finite Measures (2 P)

### U

- Uniqueness of Measures (3 P)

## Pages in category "Measures"

The following 19 pages are in this category, out of 19 total.