# Category:Measures

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This category contains results about **measures**.

### Definition 1

$\mu$ is called a **measure** on $\Sigma$ if and only if $\mu$ fulfils the following axioms:

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

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

### Definition 2

$\mu$ is called a **measure** on $\Sigma$ if and only if $\mu$ fulfils the following axioms:

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

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

\((3')\) | $:$ | \(\ds \map \mu \O \) | \(\ds = \) | \(\ds 0 \) |

## Subcategories

This category has the following 23 subcategories, out of 23 total.

### A

### B

### 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

- Null Measure (1 P)

### P

- Probability Measures (5 P)

### R

- Restricted Measures (3 P)

### S

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

### U

- Uniqueness of Measures (3 P)

## Pages in category "Measures"

The following 24 pages are in this category, out of 24 total.

### M

- Measure is Countably Subadditive
- Measure is countably subadditive
- Measure is Finitely Additive Function
- Measure is Monotone
- Measure is Strongly Additive
- Measure is Subadditive
- Measure of Empty Set is Zero
- Measure of Limit of Decreasing Sequence of Measurable Sets
- Measure of Limit of Increasing Sequence of Measurable Sets
- Measure of Set Difference with Subset
- Measure with Density is Measure