Sigma-additive set function: Difference between revisions

Revision as of 13:22, 4 October 2024

Revision as of 22:20, 18 January 2025

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===Value of empty set===

Either <math>\mu(\varnothing) = 0,</math> or <math>\mu</math> assigns <math>\infty</math> to all sets in its ___domain, or <math>\mu</math> assigns <math>- \infty</math> to all sets in its ___domain. ''Proof'': additivity implies that for every set <math>A,</math> <math>\mu(A) = \mu(A \cup \varnothing) = \mu(A) + \mu( \varnothing).</math> (it's possible in the edge case of an empty ___domain that the only choice for <math>A</math> is the empty set itself, but that still works). If <math>\mu(\varnothing) \neq 0,</math> then this equality can be satisfied only by plus or minus infinity.

===Monotonicity===