(i) (Empty set) ; 2 B.
that set as a measurable space.
Remark 1.4.13. The prex  usually denotes \countable union".
closed sets) for other instances of this prex.
closed under countable intersections as well as countable unions.
are the nite algebras and the dyadic algebras on Euclidean spaces.
algebras, but the elementary and Jordan algebras are not.
hFi is the -algebra generated by F.
cannot equal this algebra, as it is not a -algebra.
(ii) P(E) is true for all E 2 F.
(iii) If P(E) is true for some E  X, then P(XnE) is true also.
properties of Borel measurable sets (see below).
is dened to be the -algebra generated by the open subsets of X.
Elements of B[X] will be called Borel measurable.
shortly see, though, that the two -algebras are not equal.
We dened the Borel -algebra to be generated by the open sets.
Exercise 1.4.15 (Recursive description of a generated -algebra).
acollection of sets in a set X, and let !
for every countable ordinal  2 !
empty union ;), or the complement of such a union.
Remark 1.4.17. The rst uncountable ordinal !
an important role in descriptive set theory.
Borel -algebra on the diagram that you drew for Exercise 1.4.8.
ated by the union of the Borel -algebra and the null -algebra.
endow these structures with a measure.
(i) (Empty set) (;)= 0.
(E [F) = (E) +(F).
outer measure is not nitely additive on the Lebesgue algebra.
nitely additive measure on any Boolean algebra.
measure on the Lebesgue algebra (or on any of its sub-algebras).
if E 2 B and E  Y ), is also a nitely additive measure.
anitely additive measure, known as counting measure.
then (E [F)+(E \F) = (E)+(F).
are uniquely determined by .
Atriplet (X;B;), where (X;B) is a measurable space and  : B !
[0;+1] isa countably additive measure, is known as a measure space.
but the former is actually equipped with a measure.
ably additive, as is counting measure.
to a measurable subspace is again countably additive.
then c is also countably additive.
Exercise 1.4.23. Let (X;B;) be a measure space.
(i) Show that E is also B-measurable.
(ii) If there exists a B-measurable set F of nite measure (i.e.
complete if every sub-null set is a null set.
be seen from the solution to Exercise 1.4.16).
ticularly when dealing with properties that hold almost everywhere.
Exercise 1.4.26 (Completion). Let (X;B;) be a measure space.
sets that dier from a B-measurable set by a B-subnull set.
algebra on X, and let  be a measure on hAi.
exists F 2A such that (EF) < ".