Measure (mathematics) - Additivity

Additivity

Measures are required to be countably additive. However, the condition can be strengthened as follows. For any set I and any set of nonnegative ri, define:

That is, we define the sum of the to be the supremum of all the sums of finitely many of them.

A measure on is -additive if for any and any family, the following hold:

Note that the second condition is equivalent to the statement that the ideal of null sets is -complete.

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