Page 334 - DMTH401_REAL ANALYSIS

Real Analysis

Notes Intuitively, f is upper-semicontinuous at c if for x near c, f(x) is either near f(c) or less
than f(c). The function f is upper-semicontinuous if it’s upper-semicontinuous at all
points of . Prove that any upper-semicontinuous function is Lebesgue measurable.
4. We can improve Luzin’s Theorem as follows. First prove the
(i) Tietze Extension Theorem for ; named after Heinrich Tietze (1880-1964) who proved
a general result for metric spaces in 1915 [98]. Let A   be a non-empty closed set
and let f : A   be a continuous function. Prove that there is a continuous function
0
f :    such that f | = f , and if f is bounded in absolute value by a constant M,
1 1 A 0 0
then we may take f to the have the same bound. Suggestion: Show that \A is a
1
countable union of pairwise disjoint open intervals. Extend f linearly over each of
0
the open intervals to define f .
1
(ii) Using Luzin’s Theorem for n = 1, given a measurable function f : X   where X  
is measurable, prove that there is a closed set C   such that C X, m(X\C) < ,
and a continuous function g :    such that f = g on C. Moreover, if f is bounded
in absolute value by a constant M, then we may take g to have the same bound as f.

5. Here are some generalizations of Luzin’s Theorem.
(i) Let  be a -finite regular Borel measure on a topological space X, let f : X   be
measurable, and let  > 0. On “Littlewood’s First Principle(s) for regular Borel
measures,” prove that there exists a closed set C X such that m(X\C) <  and f is
continuous on C.

6. Here we present Leonida Tonelli’s (1885-1946) integral published in 1924 [100]. Let f : [a, b]
  be a bounded function, say |f| £ M for some constant M. f is said to be quasi-
continuous (q.c.) if there is a sequence of closed sets C , C , C ,...  [a, b] with lim m(C ) =
1 2 3 n n
b – a and a sequence of continuous functions f , f , f ,... where for each n, f : [a, b]  , f =
1 2 3 n
f on C , and |f | £ M.
n n n
(i) Let f : [a, b]   be bounded. Prove that f is q.c. if and only if f is measurable. To
prove the “if” statement, use Problem 6.
(ii) Let f : [a, b]   be q.c. and let {f } be a sequence of continuous functions in the
n
definition of q.c. for f. Let R(f ) denote the Riemann integral of f and prove that the
n n
limit lim R(f ) exists and its value is independent of the choice of sequence {f } in the
n n n
definition of q.c. for f. Tonelli defines the integral of f as
b
a ò f := lim R(f ).
n
n
It turns out that Tonelli’s integral is exactly the same as Lebesgue’s integral.
7. We show that the composition of two Lebesgue measurable function is not necessarily
Lebesgue measurable. Let  and M be the homeomorphism and Lebesgue measurable set,
–1
respectively. Let g =  . Show that g   is not Lebesgue measurable. Note that both  –1
M
and g are Lebesgue measurable.
8. Prove the Banach-Sierpinski Theorem, proved in 1920 by Stefan Banach (1892-1945) and
Waclaw Sierpinski (1882-1969), which states that if f :    is additive and Lebesgue
measurable, then f(x) = f(1)x for all x  . Suggestion: Observe that

 =  {x  ;|f(x)| n}.
£
n=1
Prove that for some n  , the set {x  ; |f(x)| £ n} has positive measure.

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