MATH 597: Measure Theory — Supplementary Material

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(Not graded.) Read Sections 0.1-0.3 and 0.5-0.6 in Folland’s book. Note: I expect you to have seen much but not necessarily all of this material in earlier courses. It is not necessary to know everything by heart right now. However, in order to succeed in the class, you need to be able to read mathematical material at this level of abstraction and (lack of) detail.

Approaching 597

Let $𝐴$ be an infinite (not necessarily countable) set, and $𝑓:𝐴\to ℝ$ a function. Suppose that for every integer $𝑁\ge 1$ there exist finite subsets ${𝐴}_{𝑁}^{+}\subset 𝐴$ and ${𝐴}_{𝑁}^{-}\subset 𝐴$ such that:

• (i) $|𝑓(𝛼)|\le {𝑁}^{-1}$ for all $𝛼\in 𝐴\backslash ({𝐴}_{𝑁}^{+}\cup {𝐴}_{𝑁}^{-})$;

• (ii) ${\sum }_{𝛼\in {𝐴}_{𝑁}^{+}}𝑓(𝛼)\ge 𝑁$;

• (iii) ${\sum }_{𝛼\in {𝐴}_{𝑁}^{-}}𝑓(𝛼)\le -𝑁$.

Prove that for any $𝑁\ge 1$, there exists a finite subset ${𝐵}_{𝑁}\subset 𝐴$ such that

Proof

We first take ${𝐴}_{𝑁}={𝐴}_{𝑁}^{+}\bigcup {𝐴}_{𝑁}^{-}\subset 𝐴$ s.t. $|𝑓(𝛼)|\le {𝑁}^{-1}$ for all $𝛼\in 𝐴\backslash {𝐴}_{𝑁}$, as given by the conditions.
Now we define $𝑝𝑜𝑠({𝐴}_{𝑁})≔\left\{𝛼\in {𝐴}_{𝑁}\mid 𝑓(𝛼)\ge 0\right\}$ and $𝑛𝑒𝑔({𝐴}_{𝑁})≔\left\{𝛼\in {𝐴}_{𝑁}\mid 𝑓(𝛼)<0\right\}$.
Let $𝑔𝑎𝑝≔{\sum }_{𝛼\in {𝐴}_{𝑁}}𝑓(𝛼)-597$. This is a real number since ${𝐴}_{𝑁}$ is finite.

Case 1: if $𝑔𝑎𝑝<0$, then we need to fill in more elements whose image under $𝑓$ sum up to be positive to make the sum close to 597 from below.
We then take a finite set ${𝐵}_{𝑁}^{+}\subset 𝐴$ s.t. ${\sum }_{𝛼\in {𝐵}_{𝑁}^{+}}𝑓(𝛼)\ge \lceil -𝑔𝑎𝑝+{\sum }_{𝛼\in 𝑝𝑜𝑠({𝐴}_{𝑁})}𝑓(𝛼)\rceil$.
Since ${𝐵}_{𝑁}^{+}\bigcap {𝐴}_{𝑁}\subset {𝐴}_{𝑁}$, we have

$$\sum _{𝛼\in {𝐵}_{𝑁}^{+}\bigcap {𝐴}_{𝑁}}𝑓(𝛼)\le \sum _{𝛼\in 𝑝𝑜𝑠({𝐴}_{𝑁})}𝑓(𝛼)$$

and since ${𝐵}_{𝑁}^{+}=({𝐵}_{𝑁}^{+}\backslash {𝐴}_{𝑁})\coprod ({𝐵}_{𝑁}^{+}\bigcap {𝐴}_{𝑁})$, we have

$$\sum _{𝛼\in {𝐵}_{𝑁}^{+}}𝑓(𝛼)=\sum _{𝛼\in {𝐵}_{𝑁}^{+}\backslash {𝐴}_{𝑁}}𝑓(𝛼)+\sum _{𝛼\in {𝐵}_{𝑁}^{+}\bigcap {𝐴}_{𝑁}}𝑓(𝛼)$$

By (1) and (2), it is clear that

$$\sum _{𝛼\in {𝐵}_{𝑁}^{+}\backslash {𝐴}_{𝑁}}𝑓(𝛼)\ge -𝑔𝑎𝑝$$

(3) means that the elements in ${𝐵}_{𝑁}^{+}\backslash {𝐴}_{𝑁}$ have big enough image sum to fill the gap. And by definition, for all $𝛼\in {𝐵}_{𝑁}^{+}\backslash {𝐴}_{𝑁}$, we have $|𝑓(𝛼)|\le \frac{1}{𝑁}$. This means that each element in this finite ${𝐵}_{𝑁}^{+}\backslash {𝐴}_{𝑁}$ takes up only a small portion of the sum, bounded by $1/𝑁$. Together with (3), it follows that there is some subset ${𝐵}_{𝑁}^{\prime }\subset {𝐵}_{𝑁}^{+}\backslash {𝐴}_{𝑁}$ s.t. ${\sum }_{𝛼\in {𝐵}_{𝑁}^{\prime }}𝑓(𝛼)\in [-𝑔𝑎𝑝-1/𝑁,-𝑔𝑎𝑝+1/𝑁]$. So for the finite set ${𝐴}_{𝑁}\bigcup {𝐵}_{𝑁}^{\prime }$, we have

$$\sum _{𝛼\in {𝐴}_{𝑁}\bigcup {𝐵}_{𝑁}^{\prime }}𝑓(𝛼)=\sum _{𝛼\in {𝐴}_{𝑁}}𝑓(𝛼)+\sum _{𝛼\in {𝐵}_{𝑁}^{\prime }}𝑓(𝛼)\in [597-1/𝑁,597+1/𝑁]$$

Case 2: if $𝑔𝑎𝑝>0$, then we need to fill in more elements whose image under $𝑓$ sum up to be negative to make the sum close to 597 from above.
We then take finite ${𝐵}_{𝑁}^{-}\subset 𝐴$ s.t. ${\sum }_{𝛼\in {𝐵}_{𝑁}^{-}}𝑓(𝛼)\le \lfloor -𝑔𝑎𝑝+{\sum }_{𝛼\in 𝑛𝑒𝑔({𝐴}_{𝑁})}𝑓(𝛼)\rceil$.
For the same reason as case 1, we get

$$\sum _{𝛼\in {𝐵}_{𝑁}^{-}\backslash {𝐴}_{𝑁}}𝑓(𝛼)\le -𝑔𝑎𝑝$$

And by definition, for all $𝛼\in {𝐵}_{𝑁}^{-}\backslash {𝐴}_{𝑁}$, we have $|𝑓(𝛼)|\le \frac{1}{𝑁}$. Together with (5), it follows that there is some subset ${𝐵}_{𝑁}^{\prime }\subset {𝐵}_{𝑁}^{-}\backslash {𝐴}_{𝑁}$ s.t. ${\sum }_{𝛼\in {𝐵}_{𝑁}^{\prime }}𝑓(𝛼)\in [-𝑔𝑎𝑝-1/𝑁,-𝑔𝑎𝑝+1/𝑁]$. So for the finite set ${𝐴}_{𝑁}\bigcup {𝐵}_{𝑁}^{\prime }$, we have

$$\sum _{𝛼\in {𝐴}_{𝑁}\bigcup {𝐵}_{𝑁}^{\prime }}𝑓(𝛼)=\sum _{𝛼\in {𝐴}_{𝑁}}𝑓(𝛼)+\sum _{𝛼\in {𝐵}_{𝑁}^{\prime }}𝑓(𝛼)\in [597-1/𝑁,597+1/𝑁]$$

Case 3: $𝑔𝑎𝑝=0$, then we are done.
This finishes the proof of the statement.

□

Limsup and Liminf

Let $𝑋$ be a nonempty set, and $𝐴,𝐵$ subsets of $𝑋$. Define a sequence $({𝐸}_{𝑛}{)}_{𝑛=1}^{\infty }$ of subsets of $𝑋$ by

Characterize the sets $lim\hspace{0.17em}sup{𝐸}_{𝑛}$ and $lim\hspace{0.17em}inf{𝐸}_{𝑛}$ (see §0.1 in Folland for notation).

Polynomial Convergence

Let $𝑓:{\mathbb{Z}}_{\ge 0}\times {\mathbb{Z}}_{\ge 0}\to ℝ$ be a function with the property that for every polynomial

with integer coefficients, we have that

Does it follow that $𝑓(𝑚,𝑛)\to 0$ as $𝑚,𝑛\to \infty$? In other words, given $𝜖>0$, does there exist $𝑁\ge 0$ such that $|𝑓(𝑚,𝑛)|<𝜖$ whenever $|𝑚|,\left|𝑛\right|\ge 𝑁$? Give a proof or a counterexample.

Solution

Consider this function:

$$\begin{array}{c}𝑓(𝑚,𝑛)=\{1,\text{if}𝑚=2^{𝑛}\\ 0,\text{otherwise}\end{array}$$

Let $𝑝$ be arbitrary polynomial with integer coefficients. Then there must be at most finite $𝑛$ such that $𝑝(𝑛)=2^{𝑛}$. This is guaranteed by the asymptotic behavior of polynomial and exponential function: ${lim}_{𝑛\to \infty }\frac{𝑝(𝑛)}{2^{𝑛}}=0$. So there exists some $𝑁\in \mathbb{N}$ s.t. $\frac{𝑝(𝑛)}{2^{𝑛}}<1/2$ for all $𝑛\ge 𝑁$, therefore $𝑓(𝑝(𝑛),𝑛)$ is eventually 0.
Also, there must be at most finite $𝑛$ such that $2^{𝑝(𝑛)}=𝑛$, i.e. $𝑝(𝑛)={log}_2𝑛$. This is guaranteed by the asymptotic behavior of polynomial and logarithmic function: ${lim}_{𝑛\to \infty }\frac{{log}_2𝑛}{𝑝(𝑛)}=0$. So there exists some $𝑁\in \mathbb{N}$ s.t. $\frac{{log}_2𝑛}{𝑝(𝑛)}<1/2$ for all $𝑛\ge 𝑁$, therefore $𝑓(𝑛,𝑝(𝑛))$ is eventually 0.
This confirms that ${lim}_{𝑛\to \infty }𝑓(𝑛,𝑝(𝑛))={lim}_{𝑛\to \infty }𝑓(𝑝(𝑛),𝑛)=0$ for any polynomial $𝑝$ with integer coefficients.
Then we consider the sequence $((2^{𝑛},𝑛){)}_{𝑛\in \mathbb{N}}$. For any $𝑛\in \mathbb{N}$, $𝑓((2^{𝑛},𝑛))=1$, so the sequential limit is $1$. This completes the counterexample.

1 Problem solving

Recall: Given mspace $(𝑋,\mathrm{𝒜︀},𝜇)$ 以及 $𝑓:𝑋\to ℂ$ mble, 我们可以 define distribution function:

by

Chebyshevs ineq:

for $0<𝑝<\infty$.
Today: Problem Solving

左边是 integral on $𝑋$, 右边是 integral on $ℝ$.

Write

where ${𝐴}_{𝑗}$ disjoint, ${𝑐}_1>{𝑐}_2>\cdots >{𝑐}_{𝑁}>0$ This implies:

Then

从而

Step 2: $𝑓$ general.
Use: $\exists$ simple functions ${𝑔}_{𝑛}\ge 0$ s.t. ${𝑔}_{𝑛}\nearrow ︎|𝑓|$.
MCT $⟹$

Also,

从而 MCT $⟹$

${𝜆}_{𝑓}(𝛼)=𝜇(\left\{|𝑓|>𝛼\right\})$, 以及 $\left\{|𝑓|>𝛼\right\}={\bigcup }_1^{\infty }\left\{{𝑔}_{𝑛}>𝛼\right\}$ increasing union.

2 problem solving-III

3 extra topics

3.1 Minkowski ineq for integral

3.2 convolution

我们已经证明了, for $1\le 𝑝<\infty$,

What about for $𝑝=\infty$? 答案也是 true 的, 我们需要用到 convolution 来证明.

4 Use FTC and Tonelli for series

Let ${𝑔}_{𝑘},𝑘=1,2,\dots$, be a sequence of functions that are absolutely continuous on the interval $[𝑎,𝑏]$. Suppose that there is a $𝑐\in [𝑎,𝑏]$, such that the series ${\sum }_{𝑘=1}^{\infty }{𝑔}_{𝑘}(𝑐)$ is convergent, and

(a) Show that ${\sum }_{𝑘=1}^{\infty }{𝑔}_{𝑘}(𝑥)$ is convergent for all $𝑥\in [𝑎,𝑏]$. (b) Let $𝑓(𝑥)={\sum }_{𝑘=1}^{\infty }{𝑔}_{𝑘}(𝑥)$. Show that $𝑓$ is absolutely continuous on $[𝑎,𝑏]$ and

5 Use FTC and Holder

Let $𝑓:[0,1]\to 𝑅$ be absolutely continuous, satisfy $𝑓(0)=0$ and ${𝑓}^{\prime }\in {𝐿}^2([0,1])$. Show that

exists and determine the value of this limit.                  

6 Use density of compactly supported continuous functions in a suitable space

Let $𝑓$ be a real Lebesgue measurable function on the interval $[0,1]$ such that $\parallel 𝑓\underset{\infty }{\parallel }<\infty$. Show that for any $𝜀,𝛿>0$, there is a continuous function $𝑔$ on $[0,1]$ such that $𝑚\left\{𝑥\in [0,1]:|𝑓(𝑥)-𝑔(𝑥)|>𝜀\right\}<𝛿$.                  

7 Use one of the convergence theorems

Let A be a sequence of measurable subsets of $[0,1]$ such that $inf𝑚\left({𝐴}_{𝑛}\right)>0$, where $𝑚$ stands for the Lebesgue measure. (a) Prove that there exists $𝑥\in [0,1]$ which belongs to infinitely many of the sets ${𝐴}_{𝑛}$. (b) Does there necessarily exist a point which belongs to any of the sets ${𝐴}_{𝑛}$, except finitely many?                  

8 How can we recover E from its indicator function

Let $𝐸\subset {ℝ}^1$. Show that the characteristic function ${𝜒}_{𝐸}(𝑥)$ is the limit of a sequence of continuous functions if and only if $𝐸$ is both ${𝐹}_{𝜎}$ and ${𝐺}_{𝛿}$.                 

9 be an artisan

Let $𝑓:[0,1]\to ℝ$ be a positive function of bounded variation. (a) Show that if $inf(𝑓)>0$, then the function $𝑔(𝑥)=1/𝑓(𝑥)$ is also of bounded variation on $[0,1]$. (b) Give an example of a positive function $𝑓:[0,1]\to ℝ$ of bounded variation such that $𝑔(𝑥)=1/𝑓(𝑥)$ is integrable but not of bounded variation.                 

10 Use a suitable theorem allowing you to differentiate $exp(𝑔)$ under the integral sign

Let $𝑓$ be a real Lebesgue measurable function on the interval $[0,1]$ such that $\parallel 𝑓\underset{\infty }{\parallel }<\infty$. For $𝛼\in ℝ$ define a function $𝑔(𝛼)$ by

(a) Prove that the function $𝑔(\cdot )$ is twice continuously differentiable and that ${𝑔}^{″}(𝛼)\ge 0$ for all $𝛼\in ℝ$, i.e. the function $𝑔(\cdot )$ is convex. (b) Prove that if $𝑓$ is a non-constant function, i.e. $𝑚\left\{𝑥\in [0,1]:|𝑓(𝑥)-𝑐|\ne 0\right\}>0$ for all constants $𝑐\in ℝ$, then ${𝑔}^{″}(𝛼)>0,𝛼\in ℝ$.                 

11 Use DCT

Let

Find:

12 Use Egoroff and Hölder

Let $\left\{{𝑓}_{𝑛}\right\}$ be a sequence of functions in ${𝐿}^{𝑝}\left({ℝ}^{𝑛}\right),1<𝑝<\infty$, which converge almost everywhere to a function $𝑓\in {𝐿}^{𝑝}\left({ℝ}^{𝑛}\right)$, and suppose that there is a constant $𝑀$ such that $\parallel {𝑓}_{𝑛}\underset{𝑝}{\parallel }\le 𝑀$ for all $𝑛$. Show that for every $𝑔\in {𝐿}^{𝑞}\left({ℝ}^{𝑛}\right),𝑞$ the conjugate of $𝑝$,

Is the statement true for $𝑝=1$ ? (Hint: you may want to use Egorov’s Theorem.)                 

13 Read up on HL

Let $𝑓(\cdot )$ be a locally integrable function on ${ℝ}^{𝑛}$ and $𝑀𝑓$ the corresponding Hardy-Littlewood maximal function

where $𝐵(𝑥,𝑅)$ denotes the ball centered at $𝑥$ with radius $𝑅$. a) Show that if $𝑓$ is integrable on ${ℝ}^{𝑛}$ then ${sup}_{𝜆>0}𝜆𝑚\left\{𝑥\in {ℝ}^{𝑛}:|𝑓(𝑥)|>𝜆\right\}<\infty$. b) Let $𝑓$ be the function

Show that $𝑀𝑓$ is not integrable on ${ℝ}^{𝑛}$, but ${sup}_{𝜆>0}𝜆𝑚\left\{𝑥\in {ℝ}^{𝑛}:𝑀𝑓(𝑥)>𝜆\right\}<$ $\infty$.                 

14 Use density of such functions g somewhere, and then Hölder.

Fix $1<𝑝<\infty$. Let $𝑓\in {𝐿}^{𝑝}(𝐸)$, where $𝐸$ is a measurable subset of ${ℝ}^{𝑑}$. Assume that

for all compactly supported continuous functions $𝑔:{ℝ}^{𝑑}\to ℝ$. Is $𝑓(𝑥)=0$ for almost every $𝑥$ in $𝐸$ ? If your answer is positive, prove it. Otherwise, given a counterexample.                 

15 Fubini and Tonelli

Suppose that $𝑓(𝑥),𝑥>0$, is a real valued Lebesgue measurable square integrable function. (a) Prove that for any $𝛼>0$, the inequality $2|𝑓(𝑧)||𝑓(𝑦)|\le 𝛼𝑓(𝑧{)}^2+𝑓(𝑦{)}^2/𝛼$ holds for all $𝑧,𝑦,𝛼>0$. (b) Express the double integral

as an integral over the region $\left\{0<𝑧<𝑦<\infty \right\}$. (c) Show using your work from (a) and (b) that $|𝑓(𝑧)||𝑓(𝑦)|/(𝑦+𝑧),𝑦,𝑧>0$, is integrable and

Hint: Use the inequality in (a) with $𝛼=(𝑧/𝑦{)}^{1/2}$.                 

16 Try a very nice function f first

Let $\left\{{𝑓}_{𝑛}(𝑥)\right\}$ be a sequence of continuous, strictly positive functions on $ℝ$ which converges uniformly to the function $𝑓(𝑥)$. Suppose that all the functions $\left\{{𝑓}_{𝑛}\right\},𝑓$ are integrable. Is

Justify your answer.                 

17 Use Lebesgue. Can you get the same equality for more sets E?

Let $𝑓\in {𝐿}_1([0,1],𝑑𝑥)$ be a function such that ${\int }_{𝐸}𝑓(𝑥)𝑑𝑥=0$ for any measurable set $𝐸\subset [0,1]$ of Lebesgue measure .99. Prove that $𝑓=0$ a.e.                 

18 Lebesgue

Let $𝑓\in {𝐿}^2(𝐼)$, for any finite interval $𝐼\subset ℝ$. Assume that

for all $𝑎>0$ and $𝑥\in ℝ$. Show that $|𝑓(𝑥)|\ge 1$ for a.e. $𝑥\in ℝ$.                 

19 Integration can be a trick to prove that a nonnegative function can’t be identically zero.

Let $𝑓$ and $𝑔$ be nonnegative functions in ${𝐿}^1(ℝ)$. Suppose that each function is positive on some set of positive measure. (However, there need not be a single set of positive measure where both functions are positive.) Prove that the convolution

is positive on some set of positive measure.                 

20 Check what happens on some set $\left\{𝑓<𝑐\right\}$ with $𝑐<\left|\right|𝑓|{|}_{\infty }$

Let $𝐸$ be a measurable subset of $ℝ$ such that $𝑚(𝐸)<\infty$. Let $𝑓\in {𝐿}^{\infty }(𝐸)$ with $\parallel 𝑓\underset{\infty }{\parallel }>0$. Show that

Here $\parallel 𝑓\underset{𝑛}{\parallel }≔\parallel 𝑓\underset{{𝐿}^{𝑛}(𝐸)}{\parallel },\parallel 𝑓\underset{𝑛+1}{\parallel }≔\parallel 𝑓\underset{{𝐿}^{𝑛+1}(𝐸)}{\parallel }$.                 

21 Use distribution functions

Let $𝑓:ℝ\to ℝ$ be a measurable function which has the property that

(a) Show that $|𝑓{|}^{𝑝}$ is integrable for $𝑝<3$. (b) Give an example of a function satisfying the above for which $|𝑓{|}^3$ is not integrable.