Radon-Nikodym theorem
Let μ and ν be two σ-finite measures on the same measurable space
(Ω,𝒮), such that ν≪μ
(i.e. ν is absolutely continuous
with respect to μ.)
Then there exists a measurable function
f, which is nonnegative
and finite, such that for each A∈𝒮,
ν(A)=∫Af𝑑μ. |
This function is unique (any other function satisfying these
conditions is equal to f μ-almost everywhere,) and it is called
the Radon-Nikodym derivative of ν with respect to μ,
denoted by f=dνdμ.
Remark. The theorem also holds if ν is a signed measure. Even if ν is not σ-finite the theorem holds, with the exception that f is not necessarely finite.
Some properties of the Radon-Nikodym derivative
Let ν, μ, and λ be σ-finite measures in (Ω,𝒮).
-
1.
If ν≪λ and μ≪λ, then
d(ν+μ)dλ=dνdλ+dμdλμ-almost everywhere; -
2.
If ν≪μ≪λ, then
dνdλ=dνdμdμdλμ-almost everywhere; -
3.
If μ≪λ and g is a μ-integrable function, then
∫Ωg𝑑μ=∫Ωgdμdλ𝑑λ; -
4.
If μ≪ν and ν≪μ, then
dμdν=(dνdμ)-1.
Title | Radon-Nikodym theorem |
---|---|
Canonical name | RadonNikodymTheorem |
Date of creation | 2013-03-22 13:26:15 |
Last modified on | 2013-03-22 13:26:15 |
Owner | Koro (127) |
Last modified by | Koro (127) |
Numerical id | 9 |
Author | Koro (127) |
Entry type | Theorem |
Classification | msc 28A15 |
Related topic | AbsolutelyContinuous |
Related topic | BoundedLinearFunctionalsOnLpmu |
Related topic | MartingaleProofOfTheRadonNikodymTheorem |
Related topic | BoundedLinearFunctionalsOnLinftymu |
Defines | Radon-Nikodym derivative |