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 14 subcategories, out of 14 total.
F
- Finite Signed Measures (2 P)
I
- Intersection Signed Measures (1 P)
J
- Jordan Decomposition Theorem (2 P)
- Jordan Decompositions (empty)
N
- Negative Sets (2 P)
P
- Positive Sets (1 P)
R
- Radon-Nikodym Theorem (3 P)
S
T
U
V
- Variation of Signed Measure (4 P)
Pages in category "Signed Measures"
The following 20 pages are in this category, out of 20 total.