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:


 * $\ds \sum_{n \mathop = 1}^\infty \paren {\sum_{m \mathop = 1}^\infty a_{n, m} } = \sum_{m \mathop = 1}^\infty \paren {\sum_{n \mathop = 1}^\infty a_{n, m} }$