Definition
Measure Space
A measure space is a triple $(\Omega, \mathcal{F}, \mu)$ where $(\Omega, \mathcal{F})$ is a measurable space (Definition~Sigma-Algebra) and $\mu : \mathcal{F} \to [0, \infty]$ satisfies $\mu(\emptyset) = 0$ and countable additivity.
It is $\sigma$-finite if $\Omega = \bigcup_{n=1}^\infty A_n$ with $\mu(A_n) < \infty$ for each $n$.
A probability space (Definition~Probability Space) is a measure space with $\mu(\Omega) = 1$.
Depends on
Dependency Graph
flowchart LR
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ndf8f0995["Sigma-Algebra"]
nbbb6ebd1["Probability Space"]
n0729b6b7["Measure Space"]:::current
n5373e561["Monotone Convergence Theorem"]
nace38a17["Dominated Convergence Theorem"]
n2cf1849d["Radon--Nikod\'{y}m Theorem"]
ndf8f0995 --> n0729b6b7
nbbb6ebd1 --> n0729b6b7
n0729b6b7 --> n5373e561
n0729b6b7 --> nace38a17
n0729b6b7 --> n2cf1849d
click ndf8f0995 "../objects/df8f0995.html" "_self"
click nbbb6ebd1 "../objects/bbb6ebd1.html" "_self"
click n5373e561 "../objects/5373e561.html" "_self"
click nace38a17 "../objects/ace38a17.html" "_self"
click n2cf1849d "../objects/2cf1849d.html" "_self"