For question about integration, where the theory is based on measures. So it's almost always used together with the tag [measure-theory], and its aim is to specify questions about integral, not only properties of the measure.

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Two notions of absolute continuity

If ν is a signed measure and µ a positive measure, we say that ν is absolutely continuous w.r.t. µ if µ(E) = 0 ⇒ ν(E) = 0. If |ν| is a finite measure then this ...
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1answer
31 views

Let $f \in L^1$ with $f$ differentiable at zero and $f(0)=0$. Show $\int_{-\infty}^{\infty} \frac{f(x)}{x} dx$ exists.

Is this proof good? Given the problem as stated. I first define, $$ g(x,b) = \frac{f(x)}{x}e^{ibx} $$ Which has the following property, $$ g_b(x,b) = if(x)e^{ibx} $$ And that, $$ |g_b(x,b)| = ...
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0answers
54 views

Prove g is not integrable on any interval

Q/ Let $f(x)=x^{-\frac{1}{2}}$ for $x\in(0,1)$ and 0 otherwise. Let $r_k$, k=1,2,3...be an enumeration of all rationals and set $g(x)=\sum_{k=1}^{\infty}2^{-k}f(x-r_k)$ Prove $g^2$ is finite almost ...
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15 views

Can we have $|\int _{\{f _n < f \} } (f _n - f )d \mu|<|\int _X (f _n - f )^- d \mu | $ where $f _n \to f $ (a.e.)

Can we have $|\int _{\{f _n < f \} } (f _n - f )d \mu|<|\int _X (f _n - f )^- d \mu | $ where $f _n \to f $ (a.e.) For me these two integrals are identical, but I have proof where the ...
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1answer
22 views

Continuously differentiable functions dense in $L^2[a,b]$

I read in Kolmgorov-Fomin's Элементы теории функций и функционального анализа (p. 408 here) that the set of continuously differentiable functions are dense everywhere in space $L^1[a,b]$ of Lebesgue ...
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29 views

Are these functions Lebesgue integrable?

let's consider the function $$f: [0,1] \to \mathbb{R}^+, \quad f(x) = \begin{cases} x^{-a} & x \in \mathbb{Q} \; \text{and} \; x>0\\ 0 & \text{otherwise}. \end{cases}$$ for some $a \geq ...
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3answers
51 views

Let $f \in L^1$ then prove $\lim_{b \rightarrow \infty} \int_b^{\infty} f(x) dx=0$.

So the question is as stated in the title. We are given the hint to use LDCT. Since this is homework I'm not looking for an explicit solution. I just need hints. For example, my first thoughts were ...
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1answer
27 views

Approximation of $f\in C[a,b]$ by functions constant on intervals of length $(b-a)/2^n$

I read (p. 405 here) that a continuous function on $[a,b]$ can be arbitrarily approximated according to the distance of space $L^2[a,b]$ defined by $d(f,g):=\|f-g\|_2=\int_{[a,b]}|f-g|^2d\mu$ with ...
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1answer
30 views

Convergence test for improper multiple integral

I have a function $f:\mathbb R^n \to \mathbb R$ such that $f(x)=(1+|x|)^me^{-\frac{|x|^2}{a}}$. I need to check is $$\int\limits_{\mathbb R^n}f(x)dx = \int\limits_{\mathbb R^n} ...
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2answers
25 views

Prove why this equality holds

Help in a problem about Lebesgue integration inequality If the sum is finite there is no problem , but if it is not how i can prove or show that the following happens ...
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1answer
74 views

Show a function is Lebesgue integrable

Hi I am struggling with a question but really I am struggling more with the concepts behind it so any help would be appreciated. Q/ Let $f(x)=x^{-\frac{1}{2}}$ for $x\in(0,1)$ and 0 otherwise. Let ...
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1answer
18 views

Finite sum $\sum_{r,k} p_kP_r(x_k)f(x_k)P_r(x_m)=f(x_m)$

Let $x_0,\ldots,x_n\in\mathbb{R}$ be $n+1$ arbitrary real points and $p_0,...,p_n>0$ be positive real numbers. Let $P_0,P_1,\ldots,P_n$ be polinomials such that $$\sum_{k=0}^n ...
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1answer
35 views

Proving a function is Lebesgue integrable

I need to prove that $$\frac{|x|^\alpha}{1+x^2}$$ is Lebesgue integrable for $\alpha \in [0,1)$ but I'm not sure how to do this. I first tried expanding this using the Taylor expansion to show it is ...
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0answers
52 views

Real Analysis versus Measure & Integration

I am looking at next semester's class schedule at my school, especially at a graduate course named Measure & Integration. Officially it is described as "... an introduction to the principles, ...
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65 views

Proof Riesz Representation Theorem (bounded linear functional in Lp)

I have a little problem with this proof (I'm using Royden), can you help me? Let $F$ be a bounded linear functional on $L^p$, $1 \leqslant p \leqslant \infty$. Then there is a function $ge \in L^q$ ...
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1answer
34 views

Orthogonality of Hermite functions

I would like to prove to myself that Hermite functions, defined by $\varphi_n(x)=(-1)^n e^{x^2/2}\frac{d^n e^{-x^2}}{dx^n}$, $n\in\mathbb{N}$ are an orthogonal system in $L^2(\mathbb{R})$, i.e. that, ...
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1answer
45 views

Show that $\lim_{t \to 0} \int_{\mathbb{R}^d}|f(x)-f(x-t)|dx = 0$

Let $f \in L^1(\mathbb{R}^d)$. Show that $\lim_{t \to 0} \int_{\mathbb{R}^d}|f(x)-f(x-t)|dx = 0$. What I want to do is bound $|f(x)-f(x-t)|$ above by something and then use the Lebesgue Dominated ...
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1answer
24 views

Almost everywhere (surely) properties

In Lebesgue integration, why is it so important to have properties usually true almost everywhere ? Is it because a function like $1_{\mathbb{Q}}$ is not integrable with Riemann integration ? I am not ...
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0answers
28 views

Scalar product of $L_2$ with $\mu(E):=\int_E gdx$

I read in Kolmogorov and Fomin's Элементы теории функций и функционального анализа (p. 400 here) that, if we define measure $\mu$ for $E\subset[-1,1]$ by $$\mu(E):=\int_E g(x)dx$$ where the integral ...
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50 views

Sentences about Lebesgue integrable function

In $\mathbb{R}$ with Lebesgue measure, we take $f\in L^1$ and we set $\hat{f}(t)=\int f(x) e^{ixt} dx$, for each $x$ $\ \ \ (i^2=-1)$ Show that: $\hat{f}$ is continuous $\lim_{t\rightarrow \pm ...
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1answer
66 views

Convergence as for the norm [duplicate]

If $f_n, f \in L^p, 1\leq p < +\infty$ and $f_n \rightarrow f$ almost everywhere, and $\|f_n\|_p \rightarrow \|f\|_p$, then $f_n\rightarrow f$ as for the norm. Could you give me some hints how to ...
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1answer
18 views

Restriction of a finite measure to a set on unbounded function

So I have a measure $(X,\mathscr{F},\mu)$, possibly finite, or $\sigma$-finite, or a completely general finite measure. $B\in \mathscr{F}$ is a set of finite measure. For every measure set $A\in ...
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0answers
19 views

$f$ square-summable on $X'\times X''$, $\varphi_m$ square-summable on $X'$ and $\int f\cdot\bar{\varphi}d\mu'$ square-summable on $X''$

Let $X:=X'\times X''$ be the product of measure spaces $(X',\mu')$ and $(X',\mu'')$, endowed with the Lebesge extension $\mu:=\mu'\otimes\mu''$ of product measure $\mu'\times \mu''$ defined by ...
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0answers
23 views

Is it necessary to take the inferior limit of this sequence of integrals

Suppose $A $ is a measurable set, and $\{h _n \} $ is a sequence of nonnegative simple functions such that $h _n \uparrow \chi _A $, where $\chi _A $ is the charachteristic function. I wonder why in ...
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1answer
17 views

Can I change the order of summation here?

Is it true that $\sum _{i = 1 } ^n \alpha _i \sum _{r=1 } ^{\infty } \mu(A _i \cap E _r ) = \sum _{r = 1 } ^{\infty } \sum _{i=1 } ^{n } \alpha _i\mu(A _i \cap E _r ) $ I know that I can change ...
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2answers
87 views

What is the correct definition of Area?

How is the area of a rectangle: length $\times$ breadth? We know that other areas can be derived from it. Also, the area under curves uses the area of rectangles as a basis.
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1answer
17 views

composition of measureable function with $\sqrt x$ in $L^2[0,1]$

I was wondering if my hypothesis is correct: Let $X=L^2[0,1]$, $f\in X$, $g:[0,1]\to[0,1]$ definited as $g(x)=\sqrt x$ Is $f\circ g\in L^2[0,1]$ necessarily? Thanks a lot
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1answer
62 views

Composition of measureable function with continuou function in $L^2[0,1]$

I was wondering, for general knowledge, if this claim is correct. Let $X=L^2[0,1]$, $f\in X$, $g:[0,1]\to[0,1]$ invertible. Particularly, it's image is $[0,1]$ so everything is well defined. ...
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1answer
34 views

A bounded function is Riemann integrable over [a,b] and its Rieman integral equals its Lebesgue integral

In special, M0 in Lemma 6.26 denotes the family of all step functions on real line. Dear friends, I wonder whether Sj's are needed in the proof of Lemma 6.26. Personally, I believe that S0 (S ...
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2answers
71 views

Lebesgue Integral: Convexity

Given a finite measure $\mu(\Omega)<\infty$. Consider a complex function $f\in\mathcal{L}(\rho)$. From the Riemann integral it is evident that: $$\frac{1}{\mu(\Omega)}\int_\Omega ...
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70 views

Definitions of Lebesgue integral

I know the definition, from Kolmogorov and Fomin's Элементы теории функций и функционального анализа, of Lebesgue integral of measurable function $f:X\to \mathbb{C}$ on $X,\mu(X)<\infty$ as the ...
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0answers
25 views

Continuous functions dense in $L_p(X,\mu)$ if $X$ has a special property

Let $X$ be a metric space endowed with a measure $\mu$ satisfying the following condition: all the open and closed sets of $X$ are measurable and for any measurable set $M\subset X$ and any ...
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1answer
55 views

Approximation of $f\in L_p$ with simple function $f_n\in L_p$

Let us use the definition of Lebesgue integral on $X,\mu(X)<\infty$ as the limit$$\int_X fd\mu:=\lim_{n\to\infty}\int_Xf_nd\mu=\lim_{n\to\infty}\sum_{k=1}^\infty y_{n,k}\mu(A_{n,k})$$where ...
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1answer
27 views

Convergence in $L_p$ and elsewhere

Let $\|f\|_p:=(\int_X|f|^pd\mu)^{1/p}$ and let $L_p$ be the space of (the classes of equivalence of) complex or real measurable functions such that $\int_X|f|^p d\mu<\infty$ exists. In ...
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1answer
21 views

How can I calculate the projection of a Hilbert space into a closed subspace?

I was woundering if there is an easy way to calculate the projection of a Hilbert space into a closed subspace. Obviously one could write $P:H->C$ that is given by $P(x)=d$ s.a $d=inf||x-v||$ for ...
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0answers
29 views

Is the orthogonal complement of $V=C[0,1]$ is ${0}$ in $L^2[0,1]$

I was trying to think about the orthogonal complement of $C[0,1]$ in L$^2[0,1]$. I thought that it should be $\{0\}$ but I had only little confident with my proof so I'd like to ask you if it's ...
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0answers
67 views

Counterexample to see consequence of monotone convergence theorem

Does there exist an increasing sequence of functions (fn) converging to a function f and a sequence of integrable functions (gn) such that fn>= gn for all n. And (gn) also converges to a function g ...
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2answers
32 views

Does L1 and nonnegative imply bounded almost everywhere?

Let $f:\mathbb{R}\longrightarrow\mathbb{R}$ a nonnegative function, such that $f\in L^1(\mathbb{R})$. Does this imply that $f$ is bounded almost everywhere?
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1answer
39 views

$L_2$ as a Hilbert space and $\ell_2$

I know that, if measure $\mu$, with which measure space $X$ is endowed, has a countable base, i.e. if for any measurable $M\subset X$ there exists a measurable set $A_k\in\mathscr{A}$, where ...
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1answer
54 views

Help in a problem about Lebesgue integration inequality

Let $ (X,\mathcal{S},\mu)$ be a finite measure space, let $f$ be $\mathcal{S}$-measurable and let $E_{n}:= \{x\in X :n-1\le |f(x)|<n\}$ for $n=1,2,\dots$ Show that: $$f \in ...
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1answer
30 views

$L_1\subset L_p$?

I am trying to check whether the implication $\forall p>1\quad f\in L_p(X,\mu)\Rightarrow f\in L_1(X,\mu)$ is true when $\mu(X)<\infty$. By $L_p(X,\mu)$ I mean the space of Lebesgue integrable ...
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2answers
65 views

Lebesgue integrable implies bounded a.e

I am thinking on the question: does Lesbesgue integrable implies bounded a.e? I think it does. Is Chebyshev the correct way to show it?
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1answer
27 views

Limit of integral over set measurable

If $A\subset[0,2\pi]$ is measurable, prove that $$\lim_{n\to\infty}\int_A \cos (nx)\ dx=\lim_{n\to\infty}\int_A \sin(nx) \ dx=0$$ Please, any suggestions are welcome.
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0answers
47 views

Approximating simple summable function in measure space with countable base

Let $f:X\to \mathbb{Q}+i\mathbb{Q}\subset\mathbb{C}$, $f\in L_1(X,\mu)$ be a Lebesgue-summable function taking only finitely many values $y_1,\ldots,y_n\in \mathbb{Q}+i\mathbb{Q}$ on the sets ...
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1answer
54 views

$|f+g|^p$ Lebesgue-summable if $|f|^p$ and $|g|^p$ are

I read that the Minkowski integral inequality, which I knew for Riemann integrals on $[a,b]$, holds for Lebesgue integrals in the following form:$$\forall p\geq 1\quad\quad\Bigg(\int_X |f+g|^p ...
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1answer
70 views

Coincidence of functions defining Riemann-Stieltjes integral

I read in Kolmogorov-Fomin's Элементы теории функций и функционального анализа that (p. 372 here), if the Riemann-Stieltjes integrals $$\int_a^b f(x) d\Phi_1(x)\quad\text{ and }\quad\int_a^b f(x) ...
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0answers
51 views

Applications of the Fubini-Tonelli Theorems

We are working with Lebesgue integrals right now in analysis, and in particular just proved the Fubini and Tonelli theorems. I have some questions on our latest problem set: Let $E$ be a measurable ...
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2answers
88 views

Why are some convergent Lebesgue integrals 'undefined'? [duplicate]

I sometimes read statements such as The integral $$\int_0^{\infty} dx \, \frac{\sin x}{x} $$ does not exist as a Lebesgue integral, because it is not absolutely convergent. But according to my ...
2
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1answer
75 views

Approximate Identity: Proof?

Problem Given a positive Lebesgue integrable function $j\in\mathcal{L}:j\geq0$. Consider Lebesgue integrable functions $f:\mathbb{R}\to\mathbb{R}$. Then it acts as an approximate identity: ...
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1answer
29 views

Is the supremum of two-variable measurable function always measurable

Problem : [Let $(X, \mathcal{A})$ and $(Y, \mathcal{B})$ be two measurable spaces and let $f\geq 0$ be measurable with respect to $\mathcal{A} \times \mathcal{B}$. Let $g(x)=\sup_{y\in Y} f(x, y)$ and ...