1. Measure theory
(ii) (Finite additivity) Whenever E;F 2 B are disjoint, then
(E [F) = (E) +(F).
Remark 1.4.20. The empty set axiom is needed in order to rule out
the degenerate situation in which every set (including the empty set)
has innite measure.
Example 1.4.21. Lebesgue measure m is a nitely additive measure
on theLebesgue-algebra, and henceon all sub-algebras (such as the
null algebra, the Jordan algebra, or the elementary algebra). In par-
ticular, Jordan measure and elementary measure are nitely additive
(adopting the convention that co-Jordan measurable sets have in-
nite Jordan measure, and co-elementary sets have inniteelementary
On the other hand, as we saw in previous notes, Lebesgue outer
measure is not nitely additive on the discrete algebra, and Jordan
outer measure is not nitely additive on the Lebesgue algebra.
Example 1.4.22 (Diracmeasure). Let x 2X and B be an arbitrary
Boolean algebra on X. Then the Dirac measure
at x, dened by
(E) := 1
(x), is nitely additive.
Example 1.4.23 (Zero measure). The zero measure 0 : E 7! 0 is a
nitely additive measure on any Boolean algebra.
Example1.4.24 (Linearcombinationsofmeasures). IfB isa Boolean
algebra onX, and ; : B ! [0;+1] are nitelyadditivemeasureson
B, then + : E 7! (E)+(E)is also a nitelyadditivemeasure, as
is c : E 7! c(E) for any c 2 [0;+1]. Thus, for instance, thesum
of Lebesgue measure and a Dirac measure is also a nitely additive
measure on the Lebesgue algebra (or on any of its sub-algebras).
Example1.4.25 (Restrictionofa measure). IfB isaBooleanalgebra
on X, : B ! [0;+1] is a nitely additive measure, and Y is a B-
measurable subset of X, then the restriction
Bto Y , dened by setting
(E) := (E) whenever E 2B
if E 2 B and E Y ), is also a nitely additive measure.
Example 1.4.26 (Counting measure). If B is a Boolean algebra on
X, then the function #: B ! [0;+1] dened by setting #(E) to be