(
-algebra)
A collection
of subsets of a set
is called a
-algebra
if the following conditions hold.

1.
.

2.
.

3. If
,
then
and
.

Definition

(Measurable space) The pair
is called a "measurable space" if
is a set and
is a
-algebra
of subsets of
.
A set
is called "measurable" if
.
A mapping
is called "measure" if
and
for a countable collection of disjoint sets. The triple
is called "measure space". A set
is "locally measurable" if
for
such that
.
If, for a measure space
,
we have
then
is called "probability space".

Definition

(
-finite
set). A set
is called
"
-finite"
if it is a countable union of measurable sets of finite measure.