This post elaborates on the concept of σ-algebra generated by a set, as presented in p. 17 of the book “Probability with martingales” by David Williams.
Let be a class of subsets of a set
. It is stated in the book that the σ-algebra
generated by
is the smallest σ-algebra on
such that contains
. Moreover, it is mentioned that
is the intersection of all σ-algebras on
that contain
.
Three clarifications will be made. Let be the set of all σ-algebras on
that contain
.
Firstly, it will be shown that the intersection , which is the intersection of all σ-algebras on
that contain
, is itself a σ-algebra on
that contains
. Since
for all
, it follows that
. Moreover, consider a set
. So
for all
, which means
for all
, so
. Let
be a countable collection of sets
. It follows that
for all
and all
, so
for all
, which gives
. It has thus been confirmed that
is a σ-algebra on
. As
for all
, it becomes obvious that
, so the σ-algebra
on
contains
.
Consider now those σ-algebras with
that satisfy
for all
with
. This is what is meant by “smallest σ-algebra
on
such that
” in William’s book. Assume that there are two such distinct σ-algebras
and
, i.e.
. Since
, it is then deduced that
. Similarly,
, so
, which is a contradiction. Hence, there is a single σ-algebra in
that contains
and is a subset of any other σ-algebra in
that also contains
.
It has been shown that . Moreover,
satisfies
for any
. Uniqueness has also been established, so
is the only element in
that is a subset of all
. Put in words,
is the only σ-algebra on
that contains
and is contained in any other σ-algebra
on
that satisfies
.
is called the σ-algebra generated by
and is denoted by
.