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The E=mc² Barnstar | |
message Vitalyb ( talk) 20:03, 21 February 2014 (UTC) |
Let X, Y be measure spaces with measures μ, ν respectively.
Let be a measurable function.
Then it is true that
provided one of the following criteria:
(*) For probability spaces this is automatic.
Let Ω be a measure space with a measure μ.
Let fn : Ω → ℝ be a sequence of measurable functions that converges pointwise (everywhere, or μ-almost everywhere if μ is a complete measure) to a function f : Ω → ℝ.
Then it is true that provided one of the following criteria:
μ-almost everywhere in Ω.
Note: If additionally then in L1(μ) by Scheffé’s lemma.
for some (everywhere, or μ-almost everywhere if μ is a complete measure).
Note: This also gives us in L1(μ), and .
and .
Note: This also gives us in L1(μ), and .
Let , wherein , and if ω is held constant, for all ω (or μ-almost all ω if μ is a complete measure), f is differentiable in x. Suppose F is defined in a neighborhood of 0.
Then it is true that provided one of the following criteria:
This is designed as a partition of unity.
List of canonical coordinate transformations
Let σd-1 be the uniform probability measure on the d-1-dimensional unit sphere and let κd be the volume of the d-dimensional unit ball (so that dκd is the surface area of the sphere). Then:
Corollary: If f is radial, that is: f(x) = f(|x|), then:
This may be proven using the previously-mentioned change of variables.
In particular, .
Let (Ω, P) be a probability space.
Denote by κd the volume of the d-dimensional unit ball. Then
Denote by sd-1 the surface area of the d-1-dimensional unit sphere (the boundary of the d-dimensional unit ball). Then
Let Bd(r) be the d-dimensional Euclidean ball centered at the origin with radius r. Then the following inclusion is true:
(TODO: The more general result with Hölder's inequality, inclusions of Lp spaces, etc.)
Wikipedia:Babel | ||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| ||||||||||||||||||||||||||
Search user languages |
I'm Yoni.
![]() |
The E=mc² Barnstar | |
message Vitalyb ( talk) 20:03, 21 February 2014 (UTC) |
Let X, Y be measure spaces with measures μ, ν respectively.
Let be a measurable function.
Then it is true that
provided one of the following criteria:
(*) For probability spaces this is automatic.
Let Ω be a measure space with a measure μ.
Let fn : Ω → ℝ be a sequence of measurable functions that converges pointwise (everywhere, or μ-almost everywhere if μ is a complete measure) to a function f : Ω → ℝ.
Then it is true that provided one of the following criteria:
μ-almost everywhere in Ω.
Note: If additionally then in L1(μ) by Scheffé’s lemma.
for some (everywhere, or μ-almost everywhere if μ is a complete measure).
Note: This also gives us in L1(μ), and .
and .
Note: This also gives us in L1(μ), and .
Let , wherein , and if ω is held constant, for all ω (or μ-almost all ω if μ is a complete measure), f is differentiable in x. Suppose F is defined in a neighborhood of 0.
Then it is true that provided one of the following criteria:
This is designed as a partition of unity.
List of canonical coordinate transformations
Let σd-1 be the uniform probability measure on the d-1-dimensional unit sphere and let κd be the volume of the d-dimensional unit ball (so that dκd is the surface area of the sphere). Then:
Corollary: If f is radial, that is: f(x) = f(|x|), then:
This may be proven using the previously-mentioned change of variables.
In particular, .
Let (Ω, P) be a probability space.
Denote by κd the volume of the d-dimensional unit ball. Then
Denote by sd-1 the surface area of the d-1-dimensional unit sphere (the boundary of the d-dimensional unit ball). Then
Let Bd(r) be the d-dimensional Euclidean ball centered at the origin with radius r. Then the following inclusion is true:
(TODO: The more general result with Hölder's inequality, inclusions of Lp spaces, etc.)