Summary: 5 Measure theory II
1. Charges (signed measures). Let (, A) be a -algebra. A map
: A R is called a charge, (or signed measure or -additive set func-
tion) if




j=1
Aj

=

j=1
(Aj) (5.1)
for any disjoint countable family {Aj} , Aj A .
2. The equivalent definition is to require that the series in (5.1) converges
absolutely. Indeed, the left hand side in (5.1) does not change if we replace
{Aj} by {A(j)} for any rearrangement (bijective map) : N N .
Hence the number series in the right hand side converges to the same

Source: Akhmedov, Azer - Department of Mathematics, University of California at Santa Barbara

Collections: Mathematics