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Many results in measure theory are first proven for finite measures and then extended to $\sigma$-finite ones. Here are some examples: $\pi\text{-}\lambda$ theorem approach to proving two measures are the same. Fubini-Tonelli. Lebesgue decomposition. Radon-Nikodym theorem. Other results (e.g. D...
In mathematics, more precisely in measure theory, Lebesgue's decomposition theorem states that for every two σ-finite signed measures
μ
{\displaystyle \mu }
and
ν
{\displaystyle \nu }
on a measurable space
(
Ω
,
Σ
)
,
{\displaystyle (\Omega ,\Sigma ),}
there exist two σ-finite signed measures
ν
0
{\displaystyle \nu _{0}}
and...
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