0
votes
1answer
22 views

Show that the image of a zero measure set is of zero measure

I saw a topic on the subject but I did not quite understand, and it was a bit old and I didn't want to resurrect it. I am going in the right direction, I just need a little nudge. let $f: \mathbb ...
0
votes
1answer
34 views

Support of $L^p$ functions?

I noticed something strange. If we look at a function $f \in L^p$, then this is an equivalence class. Hypothetically: $\operatorname{supp}(f) = \overline{\{f\neq 0\}}$. But this is strange, as $f$ is ...
0
votes
3answers
79 views

a continuous function, satisfying $f(α) = f(β) +f(α −β)$ for any $α, β ∈ \mathbb{R}$ [duplicate]

Hi need some help with this problem: Assume $f : \mathbb{R} → \mathbb{R}$ is a continuous function, satisfying $f(α) = f(β) +f(α −β)$ for any $α, β ∈ \mathbb{R}$, and $f(0) = 0$. Then $f(α) = α ...
2
votes
0answers
35 views

Integral inequality related to derivation

While trying to understand a proof, i have stumbled upon the following statement: Let $f \in L^p(a,b)$ be a $p$-integrable function. Then the inequality $$\liminf_{s \rightarrow t} \frac{1}{t-s} ...
1
vote
0answers
18 views

Leibniz's Derivative Rule for Integral in Measure Theory

I saw the extension of Leibniz rule for integrals for measure theory on Wiki, although I am not sure if the proposition there is correct. Besides there is no proof for it. Can anybody please introduce ...
1
vote
1answer
28 views

Weak absolute continuity of measures

I want to show that if we have a function $f \in L^p$ sucht that $||f||_p =1$. Define a new measure $\mu$ by $$\mu(A):=\int_A |f(x)|^p dm(x).$$ Then $\forall \epsilon > 0 \ \ \exists ...
9
votes
0answers
129 views

Topology of convergence in measure

Currently I am doing some measure theory(on $X=[0,1]$ with the Borel-Sigma algebra and the Lebesgue measure) and I am looking at sets$A \subset L^p$, such that for all $q \in (0,p)$, the topologies ...
0
votes
1answer
50 views

$f '$ is not Lebesgue integrable on $[-1,1]$

Let f be that function from R to R defined by f(x)= 0 if x=0 x^2 sin(1/x) if x not = 0 show that the function f' is ...
1
vote
1answer
33 views

How to determine $\sigma$-Algebra?

Let $\Omega$=$\Bbb R$ and $\mathcal R$ = {$A \subseteq \Omega : A \cap \Bbb R_+ $is Borel-set and $A \cap \Bbb R_- \in${$\emptyset, \Bbb R_-$} } What is the $\sigma$-Algebra generated by $\mathcal ...
0
votes
1answer
39 views

Need help to complete the proof about change of probability

Got a question about change of probability. $P$ and $Q$ are two probability measures on the same space $(\Omega,\Lambda)$,and let $f=\dfrac{dQ}{dP} $ denote the Randon-Nikodym derivative of $Q$ ...
2
votes
1answer
66 views

Understanding Lebesgue Integration

I have started studying Lebesgue integration and I have a few of questions regarding the Lebesgue integral: In the wikipedia entry of "Lebesgue integration" they define the Lebesgue integral as: ...
0
votes
1answer
22 views

question about the continuity of a function

I need to show that: If $f$ is continuous at $x_0$ iff $f^*(x_0)=f_*(x_0)$ where: $f^*(x_0)=\lim_{x \to x_0} \sup f(y)=\inf_{\epsilon > 0} \sup_{|y-x_0|<\epsilon}f(y)$ and $f_*(x_0)=\lim_{x ...
1
vote
1answer
63 views

Show that $\mu$ is unique and right-continuous

This problem is a little tricky, so I'd like some of your thoughts on it! Assume $\mu$ is a measure on $(\mathbb{R},\mathbb{B})$ where $\mathbb{B}$ are the Borel sets on $\mathbb{R}$. Also, ...
1
vote
1answer
38 views

$C^\omega(\Omega)\cap C^\infty_0(\Omega)$.

Let $\Omega$ denote an open connected set in $\mathbf{R}$ (AKA open interval). Is it true and how can we prove it that $C^\omega(\Omega)\cap C^\infty_0(\Omega)$ consists of the zero function alone, ...
3
votes
1answer
74 views

Why is $g_n:=\inf\{f_n,f_{n+1},…\}$ integrable if $f_n$ are?

This question is motivated by the proof of the Fatou's lemma. My text defines $g_n:=\inf\{f_n,f_{n+1},\ldots\}$ and states that it's Lebesgue integrable (each $f_n$ is). We proved that point-wise ...
7
votes
0answers
247 views

Egorov's theorem for this Lebesgue integral

I want to prove Egorov's theorem using this Lebesgue integral defined by the upper integral $$\int^*f:=\left\{\int h ; h \ge f \text{ and h upper-continuous }\right\}$$ $$\int_*f:=\left\{\int h ; h ...
2
votes
0answers
34 views

Strategies for swapping the order of integration with dependent bounds

What are the general strategies for swapping the order of integration given dependent bounds? Specifically, in $\mathbb{R}^2$, Fubini's theorem allows us the following $$ \int_{a}^b\int_{c}^d ...
6
votes
1answer
207 views

Continuous linear map

I am trying to understand how to start with this exercise. Let $\Omega \subset \mathbb{R}^n$, $n\ge 3$, open and bounded and $$ C^{1,b}(\Omega)= \{\,f\in C^1(\Omega): \text{$f$ and all its partial ...
1
vote
4answers
80 views

How many ways can you show that $ \displaystyle \int_0^1 \dfrac{\sin(\pi x)}{(1-x)^2}\ \mathrm{d}x\ $ is divergent?

How many ways can you show that this integral is divergent? $\displaystyle\int_0^1 \dfrac{\sin(\pi x)}{(1-x)^2} \,dx$ The only way I was able to show this was using a hint which was given to me that ...
2
votes
4answers
78 views

Investigate the convergence of $\sum a_n$ where $a_n = \int_0^1 \frac{x^n}{1-x}\sin(\pi x) \,dx$

Investigate the convergence of $\sum a_n$ where $a_n = \displaystyle\int_0^1 \dfrac{x^n}{1-x}\sin(\pi x) \,dx$. We have thought about using the dominated convergence theorem to find $\lim a_n$, but ...
0
votes
1answer
17 views

Which of these compact sets are possible such that they have the following measures?

I am supposed to construct compact sets $K \subset \mathbb{R}$ (if possible) that have the following properties: lambda is the Lebesgue-measure: $ \lambda (K^0) = \lambda (K)$. This is easy, just ...
5
votes
2answers
269 views

$n$th derivative of $(x^2-1)^n$

Define $R_n(x)=\dfrac{d^n}{dx^n}(x^2-1)^n$. Show that $R_n(x)$ is orthogonal to $1,x,\ldots,x^{n-1}$ in $L^2([-1,1])$. Also, what is the value of $R_n(1)$? By definition we have to show that ...
1
vote
1answer
120 views

Question regarding proof of Fatou's lemma

I'm really confused with the step enclosed in red. Can someone please be kind enough to explain to me why does it follow the part in red?
0
votes
2answers
42 views

Is sum of tail probability always less than integral of tail probability?

I'm working through some Probability and Measure Theory, and frequently we have been using the fact that for $X_i$ iid $\sum\limits_{k = 1}^{\infty} P(|X_1| > k) \leq ...
0
votes
1answer
34 views

Proving set equality

Let $\{ f_n \}$ be a sequence of functions, then I am having hard time trying to see why $$ \{ x : ( \max_{n \leq k} f_n) (x) > a \} = \bigcup_{n=1}^{k} \{ x : f_n(x) > a \} $$ these two sets ...
0
votes
1answer
207 views

Expressing intervals as a union or intersection of intervals of the form $(a,b]$

I want to express all intervals as countable union or intersection of intervals of the form $(a,b]$. I already know $$ (a,b) = \bigcup_{n} (a, b - \frac{1}{n} ]$$ $$ [a,b] = \bigcup_{n} (a + ...
0
votes
1answer
56 views

Layer-Cake for general functions

The Layer-Cake representation of a non-negative measureable function $f:\mathbb{R}^n\longrightarrow \mathbb{R}$ is given by $$f(x) = \int^{\infty}_{0} \mathbb{I}_{\{y\ \in\ ...
2
votes
1answer
90 views

Derive $\frac1n \|x\|_p^p \leq \|x\| \leq n^{p/2}\|x\|_p^p$ from Holder's inequality?

Given a vector $x = (x_1, \dotsc, x_n)\in \mathbb{C}^n$, I wanted to compare $|x_1|^p + \dotsb + |x_n|^p$ to $\|x\|^p$. I discovered that if $m=\max_i|x_i|$, we have $$m^p \leq \|x\|^p \leq ...
1
vote
3answers
33 views

Change of variables with a square

Can someone help me understand this a bit better: $\int (x-y)^2 dx = \int(y-x)^2dx$ as $(y-x)^2 = (x-y)^2$. Now, if I make the change $z = x-y$ in the one on the LHS I get: $\int z^2 dz$ as $dz ...
5
votes
2answers
1k views

Is Dirichlet function Riemann integrable?

"Dirichlet function" is meant to be the characteristic function of rational numbers on $[a,b]\subset\mathbb{R}$. On one hand, a function on $[a,b]$ is Riemann integrable if and only if it is bounded ...
4
votes
2answers
98 views

Is the validity of measuring area by approximation an assumption of calculus?

The assumption that if you subdivide an area into more and more sub intervals, the approximation gets better and better. Has this been formally proved, or is it just intuition? Thanks!
2
votes
0answers
84 views

When $ A \int_0^{\infty} e^{-\lambda t}S(t)u dt = \int_0^{\infty} e^{-\lambda t}S(t)Au dt$?

I have real Banach space $X$ and a bounded linear operator $S: X \to X$ which satisfy: 1) $S(0)u = u$ $\text{ }$ for all $u \in X$ 2) $S(t+s)=S(t)S(s)u = S(s)S(t)u $ $\quad$($ t,s \geq 0$, $u \in X ...
2
votes
1answer
482 views

Does an absolutely integrable function tend to $0$ as its argument tends to infinity?

Suppose that $f:[0,\infty)\rightarrow\mathbb{R}$ is continuous. Is it true that $$\int_{0}^\infty|f(t)|dt<\infty\Rightarrow \lim_{t\rightarrow\infty}f(t)=0?$$ If so can you provide a proof, ...
0
votes
2answers
109 views

If a continuous function from $\mathbb{R}$ to $[0,\infty)$ does not tend to zero, is its integral greater or equal than some linear function?

Consider a continuous function $f:\mathbb{R}\rightarrow[0,\infty)$ that does not tend to zero as its argument tends to infinity. Formally, there is some $\varepsilon>0$ such that there does not ...
4
votes
2answers
197 views

Interchanging closed operators and integrals

I am dealing with a problem in Evans PDE without measure theory knowledge... We have contraction semigroup $\{S_t\}_{t \geq 0}$ on real Banach space $X$, i.e family of bounded linear operators from $ ...
2
votes
1answer
136 views

Sard's theorem for absolutely continuous function

Can anyone help me proving Sard's theorem where $ f $ is a real valued absolutely continuous function on $ [a,b] $ that is to prove $ f(A) $ is measure zero where $$ A = \{x\in [a,b]\ |\ f'(x) = 0 ...
1
vote
3answers
145 views

How should I calculate the Lebesgue integral of logarithm function from zero to infinity?

Does the area under the $\ln(x)$ in $(0,+\infty)$ is measurable? If yes, how can I calculate it?
1
vote
1answer
234 views

Lebesgue measurablity of Hardy Littlewood maximal function

This question maybe embarrassingly simple, but still I wish to ask whether the Hardy Littlewood maximal function is lebesgue measurable. I know it is Borel measurable as it is lower semi continuous if ...
5
votes
3answers
187 views

$ \int^{\infty}_0 |\frac{1}{(1+x)\sqrt x}|^p ~ \mathrm dx < \infty \implies p=?$

If $ f(x) = \frac{1}{(1+x)\sqrt x} $ how to find all $ p > 0 $ such that $$ \int^{\infty}_0 |f(x)|^p dx < \infty $$ The integral is with respect to lebesgue measure. Any solution or hints would ...
2
votes
1answer
100 views

proving the sum in not integrable

This is a question from past comp exam, Let ${(g_n)_n }$ be a real continuous funtion with support in $(\frac{1}{n+1}, \frac{1}{n})$, such that $\int_0^1 g_n(t)dt =1$, for $n=1,2,3....$ Now define $$ ...
0
votes
1answer
408 views

A notation question: What does it mean $g(x) = f(x+)$

I an analysis book, I came accross to the following notation, $g(x) = f(x+)$ What does it mean? PS: I added the context (Book is Folland, Real Analysis): Theorem. Let $F: \mathbb{R} \to \mathbb{R} ...
4
votes
1answer
76 views

Average Value for an uncountable set.

Let $S$ be an uncountable set of indexed real numbers. So the same number can occur more than once in the set (although with a different index). I don't assume that there is any ordering on $S$. Is ...
3
votes
1answer
582 views

Iterated Integrals - “Counterexample” to Fubini's Theorem

Showing the iterated integrals $$\int_{[0,1]}\left[\int_{[0,1]}\frac{x^2-y^2}{(x^2+y^2)^2}dx\right]dy\quad\text{and}\quad\int_{[0,1]}\left[\int_{[0,1]}\frac{x^2-y^2}{(x^2+y^2)^2}dy\right]dx$$ are ...
3
votes
0answers
127 views

Continuity criteria for Radon-Nikodym derivative

I have been looking for results or theorems which give me regularity conditions of the Radon-Nikodym derivative, but I have not found any :( For instance, we know that if $\nu\ll\mu$ then there ...
1
vote
0answers
259 views

Inner measure and outer measure

Let $E\subset[a,b]$ be a non-mensurable set. I want to show that: $\exists F \in F_\sigma$ : $F\subset E$ , $\mu^*(F)=\mu_*(E)$ .
1
vote
0answers
117 views

Evaluation of function: derivative,integral, absolute continuity

I'm in trouble with this exercise tough I thought it was not that difficult at the begininng, now I definitely changed my mind. So I hope someone could help me to solve it. Give the function: $f:[0,1] ...
4
votes
1answer
204 views

The differentiablity of monotonic functions

Problem Suppose that $f:(a,b)\to\Bbb R$ is monotonic, and $D=\left\{\,x\;\big|\;f\textrm{ is not differentiable at }x\,\right\}$. Try to prove that for each $\eta>0$, $D$ could be covered by a ...
2
votes
1answer
46 views

Inequality between products of measures of sets

Let $X$ be a compact space and $\mu$ the Lebesgue measure, with $\mu(X)=1$. Let A and B be two subsets of $X$ with positive measure. What can I say about the relations between $\mu (A \cap B)$ and ...
0
votes
1answer
51 views

Change of Variables and independent random variables.

Suppose that we have two IID random variables, $X_1, X_2$, carried by a triple $(\Omega,\mathcal{F},P)$. While solving an exercise I ended to a point that I had to see that, $$ \iint\limits_D x_1 ...
5
votes
2answers
332 views

integral of a measure zero set

let $X$ be a finite measure space and $\{f_n\}$ be a sequence of nonnegative integrable functions, $f_n \rightarrow f\ a.e.$ on $ X$. We know that $\lim_{n \rightarrow \infty}\int_X f_n d\mu=\int_X ...