Tonelli's Theorem/Corollary
Corollary to Tonelli's Theorem
Let $\sequence {a_{n, m} }_{\tuple {n, m} \in \N^2}$ be a doubly subscripted sequence of non-negative real numbers.
Then:
- $\ds \sum_{n \mathop = 1}^\infty \paren {\sum_{m \mathop = 1}^\infty a_{n, m} } = \sum_{m \mathop = 1}^\infty \paren {\sum_{n \mathop = 1}^\infty a_{n, m} }$
Proof
Consider the measure space $\struct {\N, \map \PP \N, \mu}$ where $\mu$ is the counting measure.
From Counting Measure on Natural Numbers is Sigma-Finite, we have that $\struct {\N, \map \PP \N, \mu}$ is $\sigma$-finite.
Let $\struct {\N \times \N, \map \PP \N \otimes \map \PP \N, \mu \times \mu}$ be the product $\sigma$-algebra of $\struct {\N, \map \PP \N, \mu}$ with itself.
Define a function $f : \N^2 \to \R$ by:
- $\map f {n, m} = a_{n, m}$
for each $\tuple {n, m} \in \N^2$.
From Function Measurable with respect to Power Set, we have that $f$ is $\map \PP \N$-measurable.
We can therefore apply Tonelli's Theorem to obtain:
- $\ds \int \paren {\int \map f {n, m} \map {\rd \mu} m} \map {\rd \mu} n = \int \paren {\int \map f {n, m} \map {\rd \mu} n} \map {\rd \mu} m$
From Integral of Positive Function with respect to Counting Measure on Natural Numbers, this is equivalent to:
- $\ds \int \paren {\sum_{m \mathop = 1}^\infty \map f {n, m} } \map {\rd \mu} n = \int \paren {\sum_{n \mathop = 1}^\infty \map f {n, m} } \map {\rd \mu} m$
Applying Integral of Positive Function with respect to Counting Measure on Natural Numbers, we obtain:
- $\ds \sum_{n \mathop = 1}^\infty \paren {\sum_{m \mathop = 1}^\infty \map f {n, m} } = \sum_{m \mathop = 1}^\infty \paren {\sum_{n \mathop = 1}^\infty \map f {n, m} }$
That is:
- $\ds \sum_{n \mathop = 1}^\infty \paren {\sum_{m \mathop = 1}^\infty a_{n, m} } = \sum_{m \mathop = 1}^\infty \paren {\sum_{n \mathop = 1}^\infty a_{n, m} }$
$\blacksquare$