Page 328 - DMTH401_REAL ANALYSIS

Real Analysis

Notes 27.2 Measurability and Continuity

We saw earlier that continuity implies measurability, essentially by definition of continuity in
terms of open sets. It turns out that we can directly express measurability in terms of open sets.

Theorem 2: 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  .
Proof: To prove that (1)  (2), observe that
 
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{} =  [n, ]  f ({}) =  f [n, ].
-


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n 1 n 1
=
=
Assuming f is measurable, we have f [n, ]  S for each n and since S is a -algebra, it follows
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that f ({})  S . Also, if    is open, then by the Dyadic Cube Theorem we can write  =   n 1 I n
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=
where I  S for each n. Hence,
1
n

f ( ) =  f (I )
-
1
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n
=
n 1
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By measurability, f (I )  S for each n, so f ( )  S .
n
To prove that (2)  (3), we don’t have to worry about the preimage of , so we just have to prove
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that f (B)  S for all Borel sets B  .
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S = {A  ; f (A)  S }
f
is a -algebra. Assuming (2) we know that all open sets belong to S . Since S is a -algebra of
f f
subsets of  and B is the smallest -algebra containing the open sets, it follows that B  S .
f
Finally we prove that (3)  (1). Let a   and note that
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[a, ] = (a, )  {}  f [a, ] = f (a, )  f ({}).
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Assuming (3), we have f ({})  S and since (a, )   is open, and hence is Borel, we also have
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f (a, )  S . Thus, f (a, ]  S , so f is measurable.
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We remark that the choice of using + over – in the “f ({})  S ” parts of (2) and (3) were
arbitrary and we could have used – instead of .
Consider the second statement in the theorem, but only for real-valued functions:
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Measurability: 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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Continuity: A function f : T   is continuous if and only if f ( )  T for each open set   .
Because of this similarity, one can think about measurability as a type of generalization of
continuity. However, speaking philosophically, there are two very big differences between
n
measurable functions and continuous functions as we can see by considering X =  with Lebesgue
measure and its usual topology:
(i) There are a lot more measurable functions than continuous functions.
(ii) Measurable functions are closed under a lot more operations than continuous functions
are.
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