Page 368 - DMTH401_REAL ANALYSIS

Real Analysis

Notes (i) Let f: [0, 1]   be the characteristic function of [0, 1]  . Since f is not continuous at any
point, it is not possible to realize f as the pointwise limit of a sequence of continuous
functions from [0, 1] to , in view of [108].
(ii) Let (f ) be a sequence of continuous functions from [a, b] to  converging pointwise to a
n
function f : [a, b]  , and let Y = {x  [a, b] : f is not continuous at x}. From [108] we know
that Y is an F set of first category in [a, b]. But Y can have positive outer Lebesgue measure

by [106]. Hence f may not be Riemann integrable. Thus even the pointwise limit of a
sequence of continuous functions may not be Riemann integrable (of course, we did not
give an example).
(iii) Lebesgue integration theory is developed not just for the sake of making the characteristic
function of [0,1]   integrable. The limit theorems in Lebesgue’s theory allow us to
integrate the pointwise limit of a sequence of integrable functions, and to interchange
limit and integration, under very mild hypothesis. Moreover, the powerful tools in
Lebesgue’s theory make many proofs simpler (e.g.: the proof of the change of variable
theorem in d-dimension), and provide us with new ways of dealing with functions (e.g.: L p
spaces). Also, as we will see later, in Lebesgue’s theory we have a more satisfactory
version of the Fundamental Theorem of Calculus (describing differentiation and integration
as inverse operations of each other).

30.4 -algebras and Measurable Spaces

A d-box in  has a well-defined d-dimensional volume. We may ask whether it is possible to
d
define the notion of a d-dimensional value for all subsets of  . Of course, we would like to have
d
consistency conditions such as monotonicity and countable additivity.
d
Question: Can we have a function µ : P( )  [0, ¥] such that
d
(i) µ[A] = Vol (A) if A   is ad-box,
d
(ii) [Monotonicity] µ[A]  µ[B] for subsets A, B of  with A  B,
d
d
(iii) [Countable additivity] µ[  ¥ n 1 A ] = å ¥ n 1 µ[A ] if A ’s are pairwise disjoint subsets of  ?
=
=
n
n
n
Notes We know that the Lebesgue outer measure m * does not satisfy countable additivity.
L,d
The key observation of Lebesgue’s theory is that m * L,d will satisfy all the three conditions
stated above if we restrict m * L,d to a slightly smaller collection   ( ) by discarding
d
some pathological subsets of  . In order to describe the structure of this smaller collection
d
, it is convenient to proceed in an abstract manner, which we do below.
Definition: Let X be a nonempty set. A collection   (X) of subsets of X is said to be a -algebra
on X if the following hold:
(i) Ø, X  .
(ii) A    X\A  .

(iii) A , A ,…     ¥ A  .
=
1 2 n 1 n
If  is a -algebra on X, then (X, ) is called a measurable space.

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