Additivity

Let be a set function. is called additive or finitely additive, if whenever and are disjoint sets, then:

it follows that:

(see union, summation notation)

Value of empty set
Either , or assigns all sets in its domain

proof
Additivity implies that for every set , . If , only can satisfy this equality.

Monotonicity
If is non-negative and (see subset), then . Similarly, if is non-positive and , then .
That is, is a monotonic set function.

Set Difference
If and is defined, then

For every sequence of pairwise disjoint sets, if

holds, then is said to be countably additive or -additive

An example of a -additive function is this piecewise function:

If is a sequence of disjoint sets of real numbers, then either none of these sets contain 0, or exactly one of them does. In either case, the equality:

holds.