# uncountable sums of positive numbers

The notion of sum of a series can be generalized to sums of nonnegative
real numbers over arbitrary index sets^{}.

let $I$ be a set and let $c$ be a mapping from $I$ to the nonnegative real numbers. Then we may define the sum as follows:

$$ |

In words, we are taking the supremum over all sums over finite subsets
of the index set. This agrees with the usual notion of sum when our
set is countably infinite^{}, but generalizes this notion to uncountable
index sets.

An important fact about this generalization^{} is that the sum can only
be finite if the number of elements $i\in I$ such that ${c}_{i}>0$ is
countable^{}. To demonstrate this fact, define the sets ${s}_{n}$ (where n
is a nonnegative integer) as follows:

$${s}_{0}=\{i\in I\mid {c}_{i}\ge 1\}$$ |

when $n>0$,

$${s}_{n}=\{i\in I\mid 1/n>{c}_{i}\ge 1/(n+1)\}$$ |

If any of these sets is infinite^{}, then the sum will diverge^{} so, for
the sum to be finite, all these sets must be finite. However, if
these sets are all finite, then their union is countable. In other
words, the number of indices for which ${c}_{i}>0$ will be countable.

This notion finds use in places such as non-separable Hilbert spaces^{}.
For instance, given a vector in such a space and a complete^{}
orthonormal set, one can express the norm of the vector as the sum of
the squares of its components using this definition even when the
orthonormal set is uncountably infinite.

This discussion can also be phrased in terms of Lesbegue integration with respect to counting measure. For this point of view, please see the entry support of integrable function with respect to counting measure is countable.

Title | uncountable sums of positive numbers |
---|---|

Canonical name | UncountableSumsOfPositiveNumbers |

Date of creation | 2013-03-22 15:44:47 |

Last modified on | 2013-03-22 15:44:47 |

Owner | rspuzio (6075) |

Last modified by | rspuzio (6075) |

Numerical id | 5 |

Author | rspuzio (6075) |

Entry type | Definition |

Classification | msc 40-00 |

Related topic | SupportOfIntegrableFunctionWithRespectToCountingMeasureIsCountable |