Page 333 - DMTH401_REAL ANALYSIS

Unit 27: Measurable Functions and Littlewood’s Second Principle

27.7 Keywords Notes

Measurable Sets: A measurable space is a pair (X, S ) where X is a set and, S is a -algebra of
subsets of X. The elements of, S are called measurable sets.

Measurability Criterion: For a function f: X   , the following are equivalent:
1. f is measurable.
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2. f ({})  S and f ()  S for all open subsets   .
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3. f ({})  S and f (B)  S for all Borel sets B  .
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Measurable: A function f : X   is measurable if and only if f ( )  S for each open set   .
One cannot avoid noticing the striking resemblance to the definition of continuity. Recall that
for a topological space (T, T ), where T is the topology on a set T.
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Luzin’s Theorem: Let X   be Lebesgue measurable and let f : X   be a Lebesgue measurable
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function. Then given any  > 0, there exists a closed set C  such that C  X, m(X\C) < , and
f is continuous on C.
Borel Measurable: Any continuous real-valued function on a topological space is Borel measurable.

27.8 Review Questions

1. (a) Prove that a non-negative function f is measurable if and only if for all k   and
n
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n
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n   with 0 £ k £ 2 – 1, the sets f (k/2 , (k + 1)/2 ] and f (2 , ], are measurable.
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(b) Prove that an extended real-valued function f is measurable if and only if f ({}) and
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n
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all sets of the form f (k/2 , (k + 1)/2 ], where k   and n  , are measurable.
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(c) If {a } is any countable dense subset of , prove that f is measurable if and only if
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f ({}) and all sets of the form f (a , a ], where m, n  , are measurable.
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2. Here are some problems dealing with non-measurable functions.
(a) Find a non-Lebesgue measurable function f :    such that |f| is measurable.
(b) Find a non-Lebesgue measurable function f :    such that f is measurable.
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(c) Find two non-Lebesgue measurable functions f, g :    such that both f + g and
f  g are measurable.
3. Here are some problems dealing with measurable functions.
(a) Prove that any monotone function f :    is Lebesgue measurable.
(b) A function f :    is said to be lower-semicontinuous at a point c   if for any
 > 0 there is a  > 0 such that
|x – c|<   f(c) –  < f(x).
Intuitively, f is lower-semicontinuous at c if for x near c, f(x) is either near f(c) or
greater than f(c). The function f is lower-semicontinuous if it’s lower-semicontinuous
at all points of . (To get a feeling for lower-semicontinuity, show that the functions
 ,  , and  are lower-semicontinuous at 0.) Prove that any lower-
(0, ) (–, 0) (–, 0)  (0,)
semicontinuous function is Lebesgue measurable.
(c) A function f :    is said to be upper-semicontinuous at a point c   if for any
 > 0 there is a  > 0 such that
|x – c|<   f(x) < f(c) + .

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