Page 317 - DMTH401_REAL ANALYSIS

Unit 26: Lebesgue Measure

Proof: Easy! One can find a piecewise linear function g with the stated property. Notes
Definition: Let E  M. A function f: E   is called a simple function if there exists a , a ,..., a 
1 2 n
 and E , E ,..., E M such that
1 2 n
k
(2) f = å a 
i E i
i 1
=

Note Step function is simple,  is simple but not step function.

Proposition: Let f: [a, b]   be a simple function. For any  > 0, there is a step function : [a,
b]   such that f =  except on a set of measure less than .
Proof: Let f be given by (2), we may assume E ,E ,...,E  E. By Littlewood’s 1st Principle, there is
1 2 n
a finite union of disjoint open intervals U such that m(U  E ) < /n. Then
i i i
k n
f = å a  except on A =  (U  E ),
i U i i i
=
i 1 i 1
=
n
where m(A) < å i 1 /n = .
=
Notes One can find a continuous function with the same property. Moreover, if f satisfies
m  f  M on [a, b] then  can be chosen such that m    M (reason: replace  by (m  )
 M if necessary).

26.6 Measurable Functions

Definition: A function f: E  [-¥, ¥] is said to be measurable (or measurable on E) if E  M and

–1
f ((a, ¥])  M
for all a  .

In fact, there is a more general definition for measurability which we will not use here. The
definition goes as follows.
Definition: Let X be a measurable space and Y be a topological space. A function f: X  Y is called
measurable if f (V) is a measurable set in X for every open set V inY.
–1

Notes Simple functions, step functions, continuous functions and monotonic functions
are measurable.

Proposition: Let E  M and f : E  [–¥, ¥]. Then the following four statements are equivalent:
–1
 f ((a, ¥])  M for all a  .
–1
 f ([a, ¥])  M for all a  .
–1
 f ([–¥, a))  M for all a  .
–1
 f ([–¥, a])  M for all a  .

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