# Category:Signed Measures

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This category contains results about **Signed Measures**.

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

Let $\mu : \Sigma \to \overline \R$ be an extended real-valued function such that:

- if $\map \mu A = +\infty$ for some $A \in \Sigma$, then $\map \mu B > -\infty$ for all $B \in \Sigma$.

and:

- if $\map \mu A = -\infty$ for some $A \in \Sigma$, then $\map \mu B < +\infty$ for all $B \in \Sigma$.

We say that $\mu$ is a **signed measure** on $\struct {X, \Sigma}$ if and only if:

\((1)\) | $:$ | \(\ds \map \mu \O \) | \(\ds = \) | \(\ds 0 \) | |||||

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

## Subcategories

This category has the following 12 subcategories, out of 12 total.

### F

- Finite Signed Measures (2 P)

### I

- Intersection Signed Measures (1 P)

### J

- Jordan Decomposition Theorem (2 P)

### N

- Negative Sets (2 P)

### P

- Positive Sets (1 P)

### R

- Radon-Nikodym Theorem (3 P)

### S

### T

### U

### V

- Variation of Signed Measure (3 P)

## Pages in category "Signed Measures"

The following 17 pages are in this category, out of 17 total.