Page 369 - DMTH401_REAL ANALYSIS
P. 369
Unit 30: Riemann's and Lebesgue
Notes
Example: {Ø, X} and (X) are trivial examples of -algebras on any nonempty set X. The
d
following are some -algebras on (verify):
= {A : A or \ A is countable},
d
d
1
d
d
d
= {A : A or \ A is of first category in },
2
d
d
= {A : m * [A] = 0 or m * [ \ A] = 0}.
3 L,d L,d
d
d
d
d
= {A : [0, 1] A or [0, 1] \ A}.
4
Definition: Let X be a nonempty set and (X) be a collection of subsets of X. A -algebra on
X is said to be generated by if is the smallest -algebra on X containing . Here, exists and
is unique since is precisely the intersection of all -algebras on X containing (note that there
is at least one -algebra on X containing , namely (X)).
Definition: Let X be a metric space. Then the -algebra on X generated by the collection of all
open subsets of X is called the Borel -algebra on X, and is denoted as (X) (or just , if X is clear
from the context). The subsets of X belonging to (X) are called Borel subsets of X. For example,
open subsets, closed subsets, G subsets and F subsets of X are Borel subsets of X.
[Characterizations of the Borel -algebra on ] Consider the following collections of subsets of
d
:
d
d
= {A : A is closed},
1
= {A : A is compact},
d
2
d
= {A : A is closed d-box},
3
= {A : A is an opend-box},
d
4
d
= {A : A is ad-box},
5
d
= {A : A is an open ball},
6
–1
d
= {f (W) : f : is continuous and W is open}.
7
d
d
If is the -algebra on generated by for 1 i 7, then = ( ) for 1 i 7.
i i i
Proof: Clearly = ( ). Since (a, b) = ¥ [a + 1/n, b – 1/n] (where n is chosen so
d
=
3 2 1 n n 0 0
that a + 1/n b – 1/n ), it follows that any open d-box°is a countable union of closed d-boxes,
0 0
and therefore . Since [a,b] = ¥ (a – 1/n, b + 1/n), [a, b) = ¥ (a – 1/n, b), and (a, b] =
=
4 3 n 1 n 1
=
¥ n 1 (a, b + 1/n), we deduce that any d-box is a countable intersection of open d-boxes, and hence
=
d
= . Since any open set in can be written as a countable union of open d-boxes as well as
4 5
d
d
a countable union of open balls, we have = = ( ). Thus = ( ) for 1 i 6.
4 6 i
d
d
By the definition of continuity, we have ( ). If U is an open set different from , let
d
7
A = \ U and define f : as f(x) = dist(x, A) := inf{||x – a||: a A}. Then f is continuous, and
d
d
–1
d
A = f (0) because A is closed. Now, U = f (\{0}) and \ {0} is open in . Hence ( ) ,
–1
7
completing the proof.
Topological Remarks:
(i) If X is a separable metric space, then any base or subbase for the topology of X will
generate the Borel -algebra (X).
d
(ii) In the above characterization we used implicitly the fact that is second countable and
locally compact. If a metric space X fails to be second countable or locally compact, then
the -algebra generated by all compact subsets of X will only be a proper sub-collection of
(X). For example, try to figure out what happens for the spaces (, discrete metric) (which
is not second countable), and (, Euclidean metric) (which is not locally compact).
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