0
votes
0answers
20 views

Generalized and Lebesgue Integrable Function

I am reading a chapter about "Generalized Riemann Integrals" from Introduction to Real Analysis by Bartle & Sherbert. I have just finished reading section 10.2 which is about "Improper and ...
2
votes
2answers
69 views

Proving Lebesgue integration result

I have a Lebesgue integration question and a proposed proof. Please advise. Let $\Omega \subset \mathbb{R}^{n}$(denote the boundary as $\partial \Omega$) and consider $$\int_{\partial \Omega} vf ...
1
vote
1answer
33 views

Uniform and integral limit

Let $f_n(x)=n(\sin x)^n \cos x$. Show that the sequence of functions $f_n$ converges to $0$ uniformly on any interval of the form $[0,a]$ where $a<\pi /2$. Show that, for any continuous function ...
7
votes
2answers
52 views

Integral limit of $\sin(x/n)f(x)$

For any $f\in L^1[0,\pi]$, evaluate $n\to \infty \int^\pi_0 n$sin$(x/n)f(x)dx$ My idea is, $n$sin$(x/n)f(x)\to xf(x)$ and it seems that it is increasing sequence. I am not able to show it is ...
4
votes
1answer
52 views

Functions with every point being a Lebesgue point

For a locally integrable function $f$ a point $x$ is a Lebesgue point if the integral averages of deviations from $f(x)$ over balls centered at $x$ converge to $0$ as the balls shrink to the point. ...
0
votes
0answers
24 views

Lebesgue integration; convergence in measure [closed]

Suppose ${f_n}$ is a sequence of measurable real functions on $[0,1]$ and $\int f_n^2 \leq 1 \: \forall n$. Further, suppose $f_n \to 0$ in measure. Show $\int f_n \to 0$.
0
votes
2answers
82 views

Assume that $ f ∈ L([a, b])$ and $\int x^nf(x)dx=0$ for $n=0,1,2…$.

Assume that $ f ∈ L([a, b])$ and $\int x^nf(x)dx=0$ for $n=0,1,2...$. Prove $f=0$ a.e. Since there exist polynomials going to f almost everywhere all I would need to do is bring the limit in to ...
0
votes
3answers
49 views

If $f:[0,1]\to\mathbb{R}$ is an integrable function, prove the following:

If $f:[0,1]\to\mathbb{R}$ is an integrable function, prove the following: $\displaystyle\lim_{h\to0}\int_{[0,1]}\frac{|1+h\cdot f(t)|-1}{h}dm(t)=\int_{[0,1]}f(t)dm(t)$ I don't even know where to get ...
1
vote
2answers
37 views

The applicability of the Dominated Convergence theorem on the real line

Let $f_n(x)=\frac{1}{n}\chi_{[0,n]}(x)$, $x\in\mathbb{R}$, $n\in\mathbb{N}$ and $\chi$ is the characteristic/indicator function. Now it is clear that $f_n\rightarrow 0$, but in the text I am using it ...
2
votes
2answers
36 views

For a real valued function $f(x,y)$ on $\mathbb{R}^2$ which is in $L_2$, show that $f(x+ε,y+ε) → f(x,y)$ in $L_2$ when $ε → 0.$ [duplicate]

For a real valued function $f(x,y)$ on $\mathbb{R}^2$ which is in $L_2$, show that $f(x+ε,y+ε) → f(x,y)$ in $L_2$ when $ε → 0.$ Not sure how to go about this problem. I tried Fubini. But that ...
0
votes
0answers
13 views

For $k ∈ \mathbb{Z}$ and for $x ∈ [k/n, (k + 1)/n)$ set $g_n(x) = n\int_{k/n}^{\frac{k + 1}{n}}f(x)dx$.

Let $f ∈ L_1(\mathbb{R}).$ For $n ∈ \mathbb{N}$ define the function $g_n :\mathbb{R}→\mathbb{R}$ as follows. For $k ∈ \mathbb{Z}$ and for $x ∈ [k/n, (k + 1)/n)$ set $g_n(x) = n\int_{k/n}^{\frac{k + ...
1
vote
1answer
28 views

Finding a sequence of functions with no dominating function

Problem: Give an example of a sequence of non-negative functions $f_n$ tending to $0$ pointwise such that $\int f_n \to 0$, but there is no integrable function $g$ such that $f_n \leq g$ for all $n$. ...
2
votes
1answer
22 views

What is the proper definition of cylinder sets?

in class we defined the terminal $\sigma$-algebra for a sequence of random variables $(X_i)$ with $X_i:\Omega \rightarrow \mathbb{R}$ as $G_{\infty}:=\bigcap_i G_i$, with ...
3
votes
1answer
65 views

Show that $\lim_{x→0+ } x^{−1/2}f(x)$ exists and determine the value of this limit.

Let $f : [0,1] → \mathbb{R}$ be absolutely continuous, satisfy $f(0) = 0$ and $f′ ∈ L_2([0,1]).$ Show that $\lim_{x→0+ } x^{−1/2}f(x)$ exists and determine the value of this limit. From absolute ...
0
votes
2answers
60 views

Limit of an integral, as the measure of the region of integration approaches zero

Hi everyone: Let $f$ be a function defined on on open set $D$ of $\mathbb{R}^{N}$, $(n\geq1)$. Suppose that $(\Omega_{\varepsilon})$ is a family of measurable sets in $D$ such that ...
1
vote
1answer
62 views

Prove that $\lim_{x\to y} \frac{d(x, F )}{|x−y|} = 0$ for a.e. $y \in F$.

Let $F \subset \mathbb{R}$ be a closed set and define the distance from $x \in \mathbb{R}$ to $F$ by $d(x,F)= \inf_{y \in F} |x−y|.$ Prove that $$\lim_{x\to y} \frac{d(x, F )}{|x−y|} = 0$$ for a.e. ...
0
votes
0answers
47 views

Why is any continous function integrable?

He Everyone, For my Real Analysis course, I have a (probably very simple) problem, which I do not seem to get. Given: $f,g$ integrable functions with respect to measure $\lambda$ over the interval ...
1
vote
1answer
40 views

Let $\{f_n\}^∞_{n=1} ⊂ L_2(μ)$ be a sequence of functions such that $∥f_n∥_2 ≤ 1.$

Let $(X,A,μ)$ be a finite measure space. Let $\{f_n\}^∞_{n=1} ⊂ L_2(μ)$ be a sequence of functions such that $∥f_n∥_2 ≤ 1.$ a.) Prove that if $f_n → 0$ in measure, then $f_n → 0$ in $L_1(μ).$ b.) If ...
3
votes
2answers
70 views

Show $\lim_{n_\rightarrow \infty}\int_0^\infty ne^{-\frac{2n^2x^2}{x + 1}}dx = \infty$

As part of an analysis qual problem, I am having a hard time showing that $\lim_{n_\rightarrow \infty}\int_0^\infty ne^{-\frac{2n^2x^2}{x + 1}}dx = \infty$. Any suggestions? Thanks in advance. I ...
2
votes
0answers
26 views

Interpretation of a tail event

I am currently reading about tail events wikipedia. And I was wondering: Where does the interpretation come from that events in this sigma algebra are independent from the behaviour of any finite set ...
0
votes
1answer
45 views

Proof the that compact support is a vector space

Currently I am studying for my exam of Real Analysis, however there is one thing that I do not seem to get. Given: $$ Supp(f):=\overline{\{x\in\mathbb{R}^n:f(x)\neq 0\}} $$ the support of $f$. If the ...
0
votes
1answer
28 views

example concerning Lusin's theorem

Is there any example satisfying the following: $f$ is a measurable function on $\mathbb{R}^n$ with lebesgue measure $\lambda$. For any subset $N\subseteq\mathbb{R}^N$ with $\lambda(N)=0$, ...
4
votes
1answer
37 views

How to show convergence in distribution

Let $([0,1],B,\lambda)$ (B Borel Sigma-algebra) and $\lambda$ the Lebesgue measure. I want to show that this sequence converges in distribution. $$X_n(\omega)= \left(\begin{matrix} 1, & ...
3
votes
1answer
27 views

an argument that strengthen Lusin's theorem

Let $f$ be a measurable function on a subset $E$ of $\mathbb{R}^n$. Lusin's theorem states that for any $\epsilon>0$, there exists a measurable subset $F$ such that $F$ open in $E$, ...
0
votes
0answers
25 views

Let $f ∈ L_1((0,1)),$ and define $g : (0,1) → \mathbb{R}$ by $g(x)= \int^1_x \frac{f(t)}{t}dt$ [duplicate]

Let $f ∈ L_1((0,1)),$ and define $g : (0,1) → \mathbb{R}$ by $g(x)= \int^1_x \frac{f(t)}{t}dt$ Prove that $g ∈ L_1((0, 1)).$ Some help would be awesome. I tried doing this directly from definition ...
0
votes
1answer
22 views

Suppose there are constants $K > 0$ and $p > 1$ such that $μ\{x∈X : |f(x)|>M\}< \frac{K}{M^p}$ for all $M >0.$

Let $f$ be a measurable function on a measure space $(X,μ),$ where $μ$ is a finite measure. Suppose there are constants $K > 0$ and $p > 1$ such that $μ\{x∈X : |f(x)|>M\}< \frac{K}{M^p}$ ...
0
votes
0answers
31 views

Show that $lim_{R\rightarrow \infty}\int^R_0 \frac{\sin x}{x} dx= \frac{\pi}{2}$. [duplicate]

I came across this qualifying exam problem and wasn't sure what to do. Using techniques of real analysis (as opposed to complex analysis) show that $lim_{R\rightarrow \infty}\int^R_0 \frac{\sin x}{x} ...
1
vote
2answers
65 views

Is there an analogue of Lebesgue’s Dominated Convergence Theorem for a net $ (f_{\alpha})_{\alpha \in A} $ of measurable functions?

Is there an analogue of Lebesgue’s Dominated Convergence Theorem for a net $ (f_{\alpha})_{\alpha \in A} $ of measurable functions defined on a measure space $ (\Omega,\Sigma,\mu) $, where the index ...
0
votes
1answer
23 views

Prove that the set $A$ is measurable and find its Lebesgue measure.

Let $A ⊂ [0, 1] × [0, 1]$ be the set of points $(x, y)$ with decimal representations $x = 0.x_1x_2 ..., y = 0.y_1y_2 ...$ such that $x_ny_n = 5$ for all $n ∈ \mathbb{N}.$ Prove that the set $A$ is ...
1
vote
1answer
18 views

Let $E⊂\{(x,y)|0≤x≤1, 0≤y≤x\}, E_x =\{y|(x,y)∈E\}, E^y =\{x|(x,y)∈E\}$ and assume that $m(E_x) ≥ x^3$ for any $x ∈ [0, 1].$

Let $E⊂\{(x,y)|0≤x≤1, 0≤y≤x\}, E_x =\{y|(x,y)∈E\}, E^y =\{x|(x,y)∈E\}$ and assume that $m(E_x) ≥ x^3$ for any $x ∈ [0, 1].$ (a) Prove that there exists $y ∈ [0,1]$ such that $m(E^y) ≥ \frac{1}{4}.$ ...
1
vote
1answer
12 views

Construct a function $F : X → \mathbb{R}$ such that $F ∈ L_p(μ)$ for all $p > 1,$ but $F \notin L_1(μ).$

Let $(X,A,μ)$ be a $σ$-finite measure space with $μ(X) = ∞.$ Construct a function $F : X → \mathbb{R}$ such that $F ∈ L_p(μ)$ for all $p > 1,$ but $F \notin L_1(μ).$ I could easily do this if I ...
1
vote
3answers
104 views

Show that $\lim _{r \to 0} \|T_rf−f\|_{L_p} =0.$

I am having a hard time with the following real analysis qual problem. Any help would be awesome. Thanks. Suppose that $f \in L^p(\mathbb{R}),1\leq p< + \infty.$ Let $T_r(f)(t)=f(t−r).$ Show ...
1
vote
0answers
38 views

Let $f ∈ L_1([0,1])$ be a function such that $\int_E f(x)dx = 0$ for any measurable set $E ⊂ [0,1]$ of Lebesgue measure $0.99.$

Let $f ∈ L_1([0,1])$ be a function such that $\int_E f(x)dx = 0$ for any measurable set $E ⊂ [0,1]$ of Lebesgue measure $0.99.$ Prove that $f = 0$ a.e. Not sure how to start this question. Any ...
0
votes
0answers
35 views

Suppose $f : [0,1] → R$ satisfies $f(x) − f(y) < x − y$ for all $x,y ∈ [0,1],x > y.$

Suppose $f : [0,1] → R$ satisfies $f(x) − f(y) < x − y$ for all $x,y ∈ [0,1],x > y.$ Show that $f′$ exists almost everywhere on $[0, 1]$ or give a counterexample. Not really sure how to go ...
1
vote
1answer
32 views

One dimensional integrals in Green's theorem

I am trying to understand Green's theorem, but the problem is I don't know what is the definition of the integrals in the theorem. This is the expression that one proves to hold with some assumption ...
0
votes
0answers
28 views

Prove that for any measurable set $A ⊆ \mathbb{R}$ $\int_A g_n dm → \int_A f dm.$ [duplicate]

Let $f, g_1, g_2 . . . ∈ L_1(\mathbb{R})$ be non-negative functions. Assume that $g_n → f$ a.e. and $\int_\mathbb{R} g_n dm = \int_{\mathbb{R}} f dm$. Prove that for any measurable set $A ⊆ ...
1
vote
1answer
33 views

Prove that $\int|f − g| = \int_{-\infty}^{\infty} μ(F_t △ G_t) dt.$

Let $f$ and $g$ be integrable functions on a measure space $(X,Σ,μ).$ For each $t ∈ \mathbb{R},$ consider the sets $F_t =\{x∈X :f(x)>t\}, G_t =\{x∈X :g(x)>t\}.$ Prove that $\int|f − g| = ...
4
votes
1answer
53 views

Prove g is Lebesgue intergrable

Let $f$ be Lebesgue integrable on $(0, 1)$. For $0 < x < 1$ define g(x) = $\int_x^1t^{-1}f(t)dt$ Prove that $g$ is Lebesgue integrable on $(0, 1)$. $\int^1_0g(x)dx=\int^1_0f(x)dx.$ I am not ...
2
votes
1answer
60 views

Suppose that all the functions ${f_n},f$ are integrable. Is $lim_{n \rightarrow \infty} \int f_n(x)dx = \int f(x)dx?$

Let ${f_n(x)}$ be a sequence of continuous, strictly positive functions on $\mathbb{R}$ which converges uniformly to the function $f(x).$ Suppose that all the functions ${f_n},f$ are integrable. Is ...
0
votes
0answers
26 views

Question about a theorem concerning the continuity of integral functions

If we have $$F(t):=\int_V f(t,x)dx$$ where $V$ is some measurable subset of $\mathbb R$ and $x\mapsto f(t,x)$ is a measurable function. Moreover let $F$ be defined for all $t\in U$ a open subset of ...
1
vote
2answers
65 views

Support of a positive measure

Let $\mu$ be Lebesgue measure on $\mathbb{R}$ , $\mathcal{M}$ be Lebesgue $\sigma-$ algebra and $f\in C_{c}\left(\mathbb{R}\right)$ (continuous with compact support). Suppose $f\geq0$ over ...
0
votes
1answer
37 views

Let $E ⊂ [0,1]$ be a measurable set, $m(E) ≥ \frac{99}{100} .$ Prove that there exists $x ∈ [0,1]$

I need some help on the following real analysis past qual problem. I would appreciate some help. Let $E ⊂ [0,1]$ be a measurable set, $m(E) ≥ \frac{99}{100} .$ Prove that there exists $x ∈ [0,1]$ ...
0
votes
1answer
114 views

$\sum_{n = 1}^\infty f_n =f.$ Prove that $f′_n =f′$ a.e.

I am studying for a real analysis qualifying exam. Was hoping that there was a very slick proof for this? Thanks. Let $f_1, f_2, . . . , f : [0, 1] → \mathbb{R}$ be non-decreasing right-continuous ...
1
vote
1answer
34 views

Looking for “explicit” integrals solvable using lebesgue integration theory

I am preparing for an exam in Measure and Integration Theory (Lebesgue Integration). As far as I know my professor prefers to ask students solving explicit integrals which can be solved using the main ...
2
votes
2answers
91 views

If $f$ and $f'$ are integrable, then $f'$ has integral $0$

I have the problem Prove that if $f$ and $f'$ are Lebesgue integrable over $\mathbb{R}$, then $\int_\mathbb{R}f' = 0$, where $f'$ is defined everywhere. Honestly, not sure where to start. I had ...
2
votes
1answer
66 views

Convergence almost sure pointless?

A very common type of convergence in probability theory is 'almost sure convergence'. I don't understand why this type is used at all. In principle, we should always be able to substitute it by a ...
0
votes
1answer
33 views

Would like to compute the limit of some integral

I was working on a exercise where the goal was to compute the following limit, $$\lim_{n\rightarrow\infty}\int_{\mathbb R}e^{-|x|n}e^{-\frac{x^2}{2}}dx$$ and some tutor of mine claimed that the limit ...
3
votes
1answer
32 views

counterexample of Riemann-Lebesgue lemma for non-Borel functions

Let $f:\mathbb{R}\longrightarrow \mathbb{R}$ be a Borel measurable function. Then $$ \lim_{\lambda\to\infty}\int_{\mathbb{R}}f(x)e^{i\lambda x}d\mu(x)=0. $$ I obtain this result by showing that it is ...
1
vote
2answers
45 views

If $X,Y \subset \mathbb{R}$ are measure zero sets, how can I show that $X \times Y \subset \mathbb{R^2}$ is a measure zero set too?

If $X,Y \subset \mathbb{R}$ are measure zero sets, how can I show that $X \times Y \subset \mathbb{R^2}$ is a measure zero set too? My outline is the following: Since $X,Y$ is a measure zero set, ...
14
votes
2answers
316 views

A snappy proof of Fatou's lemma

I'm studying basic real analysis and stumbled across three big results (Fatou's lemma, Lebesgue's Dominated Convergence theorem, and the Monotone Convergence theorem) in the theory of Lebesgue ...