Page 125 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis

Notes
Then x A for some positive integer k .
o k o o
or x {x E : |f (x) – f (x)| }, k k
o n k o
which gives |f (x ) – f (x )| < , k k
n o o k o
But 0 as k .
k
Hence lim f (x ) f(x ) .
o
o
k n k
10.2 Summary

 Let E be a measurable set and R* be a set of extended real numbers. A function f : E R* is
said to be a Lebesgue measurable function on E or a measurable function on E iff the set E
(f > ) = {x E : f (x) > } = f {( , )} is a measurable subset of E R.
–1
 A property is said to hold almost everywhere (a.e.) if the set of points where it fails to hold
is a set of measure zero.
 Two functions f and g defined on the same domain E are said to be equivalent on E, written
as f ~ g on E, if f = g a.e. on E, i.e. f (x) = g (x) for all x E – E , where E E with m (E ) = 0.
1 1 1
+
–1
 f = max (f, 0) and f = max (–f, 0)
|f| = f + f –1
+
 Let A be subset of real numbers. We define the characteristic function of the set A as
A
follows:
1, if x A
(x) =
A 0, if x A
 A real valued function is called simple if it is measurable and assumes only a finite
number of values.

10.3 Keywords

Almost Everywhere (a.e.): A property is said to hold almost everywhere (a.e.) if the set of points
where it fails to hold is a set of measure zero.
Characteristic Function: Let A be subset of real numbers. We define the characteristic function
of the set A as follows:
A
1 if x A
(x) =
A 0 if x A
Egoroff’s Theorem: Let E be a measurable set with m (E) < and {f } a sequence of measurable
n
functions which converge to f a.e. on E. Then given n > 0 there is a set A E with m (A) < n such
that the sequence {f } converges to f uniformly on E – A.
n
Equivalent Functions: Two functions f and g defined on the same domain E are said to be
equivalent on E, written as f ~ g on E, if f = g a.e. on E, i.e. f (x) = g (x) for all x E – E , where E
1 1
E with m (E ) = 0.
1

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