0
votes
0answers
33 views

Prove that the function $g(·)$ is twice continuously differentiable and that $g′′(α) ≥ 0$ for all $α ∈ \mathbb{R}$, i.e.

Let $f$ be a real Lebesgue measurable function on the interval $[0, 1]$ such that $∥f∥∞ < ∞.$ For $α ∈ \mathbb{R}$ define a function $g(α)$ by $g(α) = \log \int_0^1\exp[αf(x)] dx .$ (a) Prove that ...
2
votes
0answers
32 views

“Simple” question (Lebesgue Integration) in a hard exam (proof verification)

Let $f_n\in L^1(0,1)$ and $C>0$ be such that $f_n \geq 0, f_n \rightarrow 0$ a.e., and $$\int_0^1 \max\{f_1, ..., f_n\} dx \leq C \quad \text{ for every } n.$$ Prove that $f_n \rightarrow 0$ in ...
0
votes
1answer
45 views

Find $\lim_{n \rightarrow \infty}$$\int_0^n(1 + \frac{-x}{n})^n\cos(\frac{x}{\sqrt{n}})e^{x/2}dx$

Find $\lim_{n \rightarrow \infty}$$\int_0^n(1 + \frac{-x}{n})^n\cos(\frac{x}{\sqrt{n}})e^{x/2}dx$ I want to use dominated convergence theorem obviously. However, not sure how to dominate it. ...
1
vote
0answers
40 views

Prove that $f\ast g$ is defined a.e., integrable, and such that $∥f\ast g∥_1 ≤ ∥f∥_1 · ∥g∥_1$

Let $f,g : \mathbb{R} → \mathbb{R}$ be $L_1$-functions. Set $h(x) = \int_\mathbb{R}f (x − y)g(y) \, dm(y).$ Prove that $h(x)$ is defined a.e., $h ∈ L_1(\mathbb{R})$ and $∥h∥_1 ≤ ∥f∥_1 · ∥g∥_1.$ So I ...
1
vote
0answers
60 views

Integration by substitution and separation of variables.

Let's say I want to integrate over a sphere $S^2$. Take $f \in (L^1(S^2),dS)$, then we have that $$\int_{S^2} |f| dS = \int_{S^2} |f| \sin^2 (\theta) d \theta d \phi < \infty,$$ right? Now, ...
3
votes
0answers
28 views

Let $(X,μ)$ be a measure space. Find a necessary and sufficient condition on $(X,μ)$ that $L_q(E) ⊂ L_p(E)$ for all $1 ≤ p < q ≤ ∞.$

Let $(X,μ)$ be a measure space. Find a necessary and sufficient condition on $(X,μ)$ that $L_q(E) ⊂ L_p(E)$ for all $1 ≤ p < q ≤ ∞.$ I want to say that the condition is that $E$ is finite. This ...
2
votes
0answers
35 views

Show that $\int_E f (x, y) dx$ is differentiable with respect to $y$ and $\frac{d}{dy}\int_E f(x,y)dx=\int_E \frac{d}{dy}f(x,y)dx.$

Assume that $f = f(x,y)$ is a function defined on $E × (a,b).$ For each fixed $y ∈ (a,b),$ $f$ is integrable with respect to $x$ on $E$, and for each fixed $x ∈ E$, $f$ is differentiable with respect ...
2
votes
1answer
40 views

Are these two expression square integrable?

I have two expressions (let's call them functions $f,g$) on $[0,1]$, where I want to find out whether they are square-integrable or better: for which $m \in \mathbb{Z}$ they are square-integrable ( ...
3
votes
1answer
24 views

Prove this function is absolute continuous and Lipschitz of order $\alpha$

Let $1≤p≤\infty$ and $ f \in L^{p}(a,b)$ such as there is a function $g \in L^{p}(a,b)$ that for all $\phi \in C^{1}(a,b)$ (and continuos in $[a,b]$) with $\phi(a)=\phi(b)=0$ we have: $\int_{a}^{b} ...
1
vote
0answers
26 views

Spherical coordinates and Lebesgue integral

I found a textbook that says that $r \in [0,\infty), \theta \in [0,\pi], \phi \in [-\pi,\pi]$ are the spherical coordinates. Then he goes on with: This corresponds to a unitary map $L^2(\mathbb{R}^3) ...
2
votes
1answer
34 views

What is the definition of this set of absolutely continuous function

I know that $$AC(a,b):=\left\{f \in C(a,b)|f(x) = f(c)+\int_c^x g(t) d \lambda(t),c \in (a,b), g \in L^1_{\text{loc}}(a,b)\right\}$$ $$AC[a,b]:=\left\{f \in C[a,b]|f(x) = f(c)+\int_a^x g(t) d ...
5
votes
1answer
38 views

Let $f_n:\mathbb{R}\rightarrow [0, 1]$ be functions such that $\sup_{x \in \mathbb{R}}f_n(x) = 1/n$ and $||f||_1 = 1$.

Let $f_n:\mathbb{R}\rightarrow [0, 1]$ be functions such that $\sup_{x \in \mathbb{R}}f_n(x) = 1/n$ and $||f_n||_1 = 1$. Set $F(x) = \sup_{n \in \mathbb{N}}f_n(x)$. Prove that $\int_\mathbb{R}F(x)dx ...
0
votes
0answers
35 views

Absolute continuous function and Jensen inequality

Let $f:[0,1]\rightarrow \mathbb{R}$ be an absolute continuous function, such as $f' \in L^4(0,1)$ and $f(0)=0$. Find $r<0$, such as, for all $\alpha \geq r,$ $lim_{x\rightarrow 0^{+}} ...
0
votes
1answer
31 views

A Fubini-Tonelli's Theorem Problem

Let $E \subseteq \mathbb{R^n}$ a measurable set, such as for almost every $x \in \mathbb{R^n}$ we have $|E \triangle (E+x)|=0$ (Where $\triangle$ means simetric difference between two sets and ...
1
vote
1answer
36 views

Approximation functions in $L^{1}$ by indicator functions of dyadic cubes

Let $\mu$ be a finite positive regular Borel measure on $\mathbb{R}^{d}$ and let $S$ be the family of finite unions of squares of the form $\{a_{1}2^{n} \leq x_{1} \leq (a_{1} + 1)2^{n}, \ldots, ...
2
votes
1answer
35 views

Changing the order of integration (for Lebesgue-Stieltjes integral and Riemann integral)

Do the Lebesgue-Stieltjes integral and the Riemann integral have the same rules about the change of order of integration? I mean I know how to deal with Riemann integral, but I'm not sure if I can ...
1
vote
2answers
62 views

$f$ unbounded in $\mathbb{R}$ implies it cannot be in $L^2(\mathbb{R})$.

Intuitively , I feel this must be true. I'm looking for a rigorous proof. So,I would like to confirm that there are no counter examples to begin with.
5
votes
1answer
63 views

$\int_0^1f(x)dx = 2, \int_0^1g(x)dx = 1, \text{and} \int_0^1[f(x)]^2 dx ≤ C$ for some constant $C > 4.$

Suppose $f$ and $g$ are nonnegative measurable functions on the interval $[0,1],$ with the properties $$\int_0^1 f(x)\,dx = 2, \int_0^1g(x)\,dx = 1, \text{ and }\int_0^1[f(x)]^2 dx \le C$$ for some ...
0
votes
2answers
25 views

Show that $\int_X gdν=\int_X gfdμ$ for all $g∈L_1(ν).$

Let $μ$ and $ν$ be finite (positive) measures on a measurable space $(X, M),$ and suppose that $ν(E)=\int_E fdμ$, for all $E∈M,$ $E$ where $f$ is some function in $L_1(μ).$ Show that $\int_X ...
2
votes
2answers
73 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
54 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
56 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. ...
2
votes
3answers
119 views
+50

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
56 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
38 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
38 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
15 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
29 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
23 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
71 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
61 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
63 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
44 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 ...
3
votes
0answers
29 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 ...
1
vote
1answer
49 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: $$ \mathrm{Supp}(f):=\overline{\{x\in\mathbb{R}^n:f(x)\neq 0\}} $$ the support of ...
0
votes
1answer
30 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
38 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
32 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
70 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
24 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
13 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
105 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
39 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 $0.99.$ [duplicate]

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 ...