3
votes
2answers
67 views

Without Lebesgue

Everyone knows following problem. Let $f$ be positive function on $[0,1]$ and there exist $I = \int_{0}^{1}f(x)dx$. Prove that $I>0$. (recall that there are only two cases: $I=0$ or $I>0$. NOT ...
5
votes
1answer
51 views

$xf''(x) , xf', f \in L^{2}$ is $f' \in L^{1}$?

I am stuck on the following problem. I have a function $f$ such that $f$ is bounded on $(0,1)$, $xf'(x)$ is bounded on $(0,1)$, $f \in L^{2}(0,1)$, $xf' \in L^{2}(0,1)$, and $xf'' \in L^{2}(0,1)$. ...
1
vote
1answer
28 views

variation of a function over countable intervals

Let $f$ be a function of bounded variation on $[0,1]$. Let $\{[a_n,b_n]\}_{n=1}^\infty$ such that $(a_n,b_n)$ are pairwise disjoint and $\cup_{n=1}^\infty [a_n,b_n]=[0,1]$. (for example, $[1/2, 1], ...
2
votes
1answer
30 views

Does the limit $\lim_{k\to\infty}\int|\cos kx |f(x) d\lambda(x)$ always exist?

Let $f$ be a Lebesgue integable function. Does the limit $$\lim_{k\to\infty}\int|\cos kx |f(x) d\lambda(x)$$ always exist?
1
vote
2answers
38 views

Jordan measure zero discontinuities a necessary condition for integrability

The following theorem is well known: Theorem: A function $f: [a,b] \to \mathbb R$ is Riemann integrable if and only if its set of discontinuities has Lebesgue measure zero. Now if we change ...
0
votes
0answers
33 views

find an adjoint operator for operator N [closed]

I read this in a paper how to find its adjoint?
2
votes
2answers
36 views

The set of $x$ where a sequence convergences in terms of set operations

I'm befuddled by this. Suppose $f:\mathbb{R}\to\mathbb{R}$, $f_n:\mathbb{R}\to\mathbb{R}$, $n=1,2,\dots$, and consider the set $$\bigcap_{k\geq 1}\bigcup_{p\geq 1}\bigcap_{m\geq p}\{x\in\mathbb{R} \ ...
0
votes
1answer
24 views

the area of the image under a specific holomorphic function of the unit disk

Let $f(z)=z^3+\frac{z^2}{2}$. Let $D$ be the unit disk in $\mathbb{C}$. How to compute $$ Area(f(D))? $$ In the case that $f:D\to \mathbb{C}$ is injective, \begin{align*} Area(f(D))&= \int_D ...
0
votes
1answer
36 views

bounded function continuous except for a set of measure zero

Let $f$ be a bounded real function on $\mathbb{R}^n$ and $P$ be a subset of $\mathbb{R}^n$ with Lebesgue measure zero. If $f$ is continuous on $P^c$, then $f$ is Riemann integrable. Is it true? my ...
2
votes
1answer
37 views

Convex , then also Measurable

I was reading about Jensen's inequality and noticed that don't require $\phi$ to be measurable here: Wikipedia link. Therefore, I guess that being convex implies being measurable somehow, but I have ...
0
votes
1answer
37 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
38 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
88 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
38 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
39 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
32 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 ...
12
votes
2answers
225 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
62 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
36 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
46 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 $Q$ ...
1
vote
1answer
92 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
23 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
64 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
42 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
265 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
52 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
221 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
90 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
18 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
328 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
122 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
45 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
35 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
210 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
60 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
34 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
99 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
85 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
552 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
114 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
208 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
145 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
152 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
253 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
188 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 $$ ...