Difference between revisions of "User:Boris Tsirelson/sandbox2"
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It is possible to treat absolutely convergent series as a special case of [[Lebesgue integral]]s. 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 | It is possible to treat absolutely convergent series as a special case of [[Lebesgue integral]]s. 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 | ||
| − | * | + | * a function $a$ is integrable if and only if the series $\sum a_n$ converges absolutely, and |
* in this case $\int_{\{0,1,2,\dots\}} a(n)\,\mu(\rd n) = \sum_{n=0}^\infty a_n$. | * 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. | The same holds when $a_n$ are complex numbers or elements of a Banach space. | ||
Revision as of 18:21, 21 July 2012
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
- a 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 order between points. The claims about multiple and iterated series become special cases of Fubini theorem.
On the other hand, absolutely convergent series may be used when constructing Lebesgue integral.
Boris Tsirelson/sandbox2. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Boris_Tsirelson/sandbox2&oldid=27162