Pasha Zusmanovich
TMAIN Measure Theory and Integration, winter semester 2018/2019
A very brief synopsis:
Exam:
Set 1
Set 2
Set 3
Set 4
Set 5
LITERATURE
-
M. Brokate and G. Kersting, Measure and Integral, Birkhäuser, 2015.
(a primary source)
- F.E. Burk, A Garden of Integrals, MAA, 2007.
-
S. Cheng, A Short Course on the Lebesgue Integral and Measure Theory,
Manuscript, 2004.
-
P.R. Halmos,
Measure Theory, Van Nostrand, N.Y./Macmillan London, 1950;
reprinted by Springer, 1974.
-
V. Jarník,
Integralní počet (II), Academia, Praha, 1984 (in Czech).
-
S.G. Krantz, The Integral: A Crux for Analysis,
Morgan & Claypool Publ., 2011.
-
B. Makarov and A. Podkorytov,
Real Analysis: Measures, Integrals and Applications, Springer, 2013.
-
S. Ovchinnikov,
Measure, Integral, Derivative. A Course on Lebesgue's Theory,
Springer, 2013.
-
T. Tao,
An Introduction to Measure Theory, AMS, 2011.
For an historical perspective, see
-
D.M. Bressoud, A Radical Approach to Lebesgue's Theory of Integration,
Cambridge Univ. Press, 2008.
All the books are available in electronic form in multiple places.
HOMEWORKS
Homeworks could be written in English or Czech, and submitted either in paper
form or by email. If you write and not type, please be sure
your handwriting is absolutely clear. Please do not forget to specify your name
(if submitting on paper), the name of the course, and the number of the
homework.
All the points earned by doing homeworks (maximum 15-20
points for the whole semester) will be added to your score at the final exam.
Homework 1 (0.5 points)
Is it true that \(A \backslash (B \backslash C) = (A \backslash B) \cup C\)
for any 3 sets \(A,B,C\)?
Homework 2 (2 points) (Ovchinnikov, Ex. 1.36)
Which of the following sets are open, or closed in the Euclidean (standard)
topology on \(\mathbb R\)?
- \(\bigcup_{k=1}^\infty (\frac{1}{2k},\frac{1}{2k-1})\);
- \(\bigcup_{k=1}^\infty [\frac{1}{2k},\frac{1}{2k-1}]\);
- All rational numbers with denominators that are less than \(10^6\);
- All rational numbers with denominators that are powers of 2.
Homework 3 (1 point)
Prove that the following sets form an algebra. Do they form a
\(\sigma\)-algebra?
-
The set consisting of all countable subsets, and all complements to
countable subsets of a set X.
-
The set consisting of all bounded subsets, and all complements to bounded
subsets of \(\mathbb R^2\).
Homework 4 (2 points)
Generalize example 2) from Exercise 2.8 (see
synopsis) of a set which is simultaneously a
\(\sigma\)-algebra and a topology, to the case of several proper subsets \(A_1, \dots, A_n\ \subset X\).
Homework 5 (0.5 points)
Find realization of the Galois field
\(GF(4)\)
as an algebra of sets.
Homework 6 (1 point)
(Redo of an exercise we failed to do properly in the class).
Let \(X\) be a set and \(A \subseteq X\) its subset. Describe
\(\sigma\)-algebra on \(X\) generated by \(P(A)\).
Homework 7 (2 points)
Give an example of:
- A non-measurable map.
-
A measurable but not continuous map. (Measurability is understood in the sense
of \(\sigma\)-algebra generated by topology).
(Hint: think about finite sets, or other "toy" examples).
Homework 8 (1 point)
Give an example of a non-negative additive function which is not a measure.
Homework 9 (0.5 points)
Prove the equivalence (i) \(\Leftrightarrow\) (iii) in Theorem 5.4
(see synopsis).
Homework 10 (1.5 points)
-
We have proved in the class that if \(f, g\) are two measurable functions from a
measurable space \(X\) to \(\overline{\mathbb R}\) then the set
\(\{x \in X \>|\> f(x) = g(x)\}\) is measurable. Does this remain true if
\(f, g\) are measurable functions from \(X\) to not necessary
\(\overline{\mathbb R}\), but to (another) arbitrary measurable space?
-
Prove that if \(f, g\) are two measurable functions from a measurable space
\(X\) to \(\overline{\mathbb R}\), then the set
\(\{x \in X \>|\> f(x) \le g(x)\}\) is measurable.
Homework 11 (many points)
In the class, we have (incorrectly) defined a completion of a \(\sigma\)-algebra
\(\Sigma\) with a measure \(\mu\) on it, as
\[
\widehat \Sigma =
\{ A \subseteq S \>|\> B \subseteq A \subseteq C \text{ for some } B,C \in \Sigma
\text{ such that } \mu(B) = \mu(C) \}
\]
Will be the resulting \(\widehat \Sigma\) always a \(\sigma\)-algebra?
Homework 12 (1 point)
Assuming existence of non-measurable sets (with respect to Borel measure) in
\(\mathbb R\), prove existence of non-measurable sets in \(\mathbb R^n\),
\(n>1\).
Homework 13 (2 points)
(Brokate-Kersting, Exercise 4.7)
Let \(\mu\) be a measure on a measure space \(S\), and let \(I\) be a map, which
assigns to every measurable function
\(f: S \to \mathbb R_{\ge 0} \cup \{+\infty\}\), a number
\(I(f) \in \mathbb R_{\ge 0} \cup \{+\infty\}\), and which satisfies the
following properties:
-
If \(f, g \ge 0\) are measurable, \(\lambda, \mu \in \mathbb R_{\ge 0}\), then
\(I(\lambda f + \mu g) = \lambda I(f) + \mu I(g)\).
-
If \(0 \le f_1 \le f_2 \le \dots\) are measurable, then
\(I (\sup_n f_n) = \sup_n I(f_n)\).
-
\(I(\mathbf 1_A) = \mu(A)\) for any measurable set \(A \subseteq S\).
Prove that \(I(f) = \int f d \mu\) for any measurable function \(f \ge 0\).
Homework 14 (1 point)
Suppose \(f\) is a Lebesgue integrable function. Prove that
\[
\lim_{x\to\infty} \int_x^{x+1} f \operatorname{d}t = 0 .
\]
Homework 15 (0.5 points)
Suppose \(f\) is a Lebesgue integrable function, and \(a,\lambda > 0\). Prove that
\[
\int_0^a f(x) \operatorname{d}x =
\lambda \int_0^{\frac{a}{\lambda}} f(\lambda x) \operatorname{d}x .
\]
Created: Sun Sep 25 2016
Last modified: Thu Dec 02 17:38:21 Central Europe Standard Time 2021