Page 332 - DMTH401_REAL ANALYSIS

Real Analysis

Notes Self Assessment

Fill in the blanks:
1. 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 .......................
2. For Lebesgue measure, continuity implies .......................

3. A function f : T   is continuous if and only if f ( )  T for each open set .......................
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4. Measurable functions are similar to ......................., but there are more of them and they are
more robust.
5. ....................... principle shows exactly how “similar” measurable functions are to continuous
functions.
6. Any continuous real-valued function on a topological space is .......................
7. If f :    is Borel measurable and g : X   is measurable, where X is an arbitrary
measurable space, then the composition, ....................... is measurable.

27.6 Summary

 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. Recall that a measure space is a triple (X, S , )
where  is a measure on S ; if we leave out the measure we have a measurable space.
In the discussion at the beginning of this chapter we saw that in order to define the integral
of a function f : X   , we needed to require that

1
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f (I)  S for each I  S and f [a, ]  S for each a  .
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 If these properties hold, we say that f is measurable. It turns out that we can omit the first
condition because it follows from the second. Indeed, since
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f [a, b] = f [a, ]\f [b, ].
 There are three principles, roughly expressible in the following terms: Every [finite
Lebesgue] measurable set is nearly a finite union of intervals; every measurable function
is nearly continuous; every convergent sequence of measurable functions is nearly
uniformly convergent.

 The third principle is contained in Egorov’s theorem, which we’ll get to in the next section.
The second principle comes from Luzin’s Theorem, named after Nikolai Nikolaevich
Luzin (1883-1950) who proved it in 1912 [70], and this theorem makes precise Littlewood’s
comment that any Lebesgue measurable function is “nearly continuous”.
 Any continuous real-valued function on a topological space is Borel measurable.

n
 The proof of this proposition follows word-for-word the  case in Example, so we omit
its proof. A nice thing about Borel measurability is that it behaves well under composition.
 If f :    is Borel measurable and g : X   is measurable, where X is an arbitrary
measurable space, then the composition,
f o g : X  

is measurable.

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