section_path: Generalizations/Summations over arbitrary index sets/Families of non-negative numbers
section_name: Families of non-negative numbers
paragraph_text: Families of non-negative numbers
  
  When summing a family { a i: i ∈ I } {\displaystyle \left\{a_{i}:i\in I\right\}} of non-negative real numbers, define
  
  ∑ i ∈ I a i = sup { ∑ i ∈ A a i: A ⊆ I, A finite } ∈. {\displaystyle \sum _{i\in I}a_{i}=\sup \left\{\sum _{i\in A}a_{i}\,:A\subseteq I,A{\text{ finite}}\right\}\in.}
  
  When the supremum is finite then the set of i ∈ I {\displaystyle i\in I} such that a i > 0 {\displaystyle a_{i}>0} is countable. Indeed, for every n ≥ 1, {\displaystyle n\geq 1,} the cardinality | A n | {\displaystyle \left|A_{n}\right|} of the set A n = { i ∈ I: a i > 1 / n } {\displaystyle A_{n}=\left\{i\in I:a_{i}>1/n\right\}} is finite because
  
  1 n | A n | = ∑ i ∈ A n 1 n ≤ ∑ i ∈ A n a i ≤ ∑ i ∈ I a i < ∞. {\displaystyle {\frac {1}{n}}\,\left|A_{n}\right|=\sum _{i\in A_{n}}{\frac {1}{n}}\leq \sum _{i\in A_{n}}a_{i}\leq \sum _{i\in I}a_{i}<\infty.}
  
  If I {\displaystyle I} is countably infinite and enumerated as I = { i 0, i 1, … } {\displaystyle I=\left\{i_{0},i_{1},\ldots \right\}} then the above defined sum satisfies
  
  ∑ i ∈ I a i = ∑ k = 0 + ∞ a i k, {\displaystyle \sum _{i\in I}a_{i}=\sum _{k=0}^{+\infty }a_{i_{k}},} provided the value ∞ {\displaystyle \infty } is allowed for the sum of the series.
  
  Any sum over non-negative reals can be understood as the integral of a non-negative function with respect to the counting measure, which accounts for the many similarities between the two constructions.