User:Boris Tsirelson/sandbox2

Relations to Lebesgue integral

It is possible to treat absolutely convergent series as a special case of Lebesgue integrals. To this end, the countable set $\{0,1,2,\dots\}$ is treated as a measure space; all subsets are measurable, and the counting measure $\mu$ is used: $\mu(A)$ is the number of points in $A$ ($\infty$ if $A$ is infinite). A sequence $(a_n)$ of real numbers is just a function $a:\{0,1,2,\dots\}\to\R$ (measurable, since everything is measurable on this discrete space). It is easy to see that

• the function $a$ is integrable if and only if the series $\sum a_n$ converges absolutely, and
• in this case $\int_{\{0,1,2,\dots\}} a(n)\,\mu(\rd n) = \sum_{n=0}^\infty a_n$.

The same holds when $a_n$ are complex numbers or elements of a Banach space. In this formulation, the order of terms is evidently irrelevant, since the notion of a measure space does not stipulate any ordering of points.