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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3
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3answers
121 views

triple integral $\iiint e^{-x^2-y^2-z^2+xy+yz+xz} \,dx\,dy \,dz$

I need to solve this triple integral $$\iiint e^{-x^2-y^2-z^2+xy+yz+xz} \,dx\,dy \,dz$$ where $V$ is all $\Bbb R^3$ I've spent on this task a few hours, ok firstly ...
0
votes
1answer
30 views

If $f_n \to g$ in $L_2$, and that $f_n = \mathbb{E}(f \mid \mathcal{F}_n)$, show if $f$ continuous, then $g(x)=f(x)$ for all $x$ with probability 1?

Let $\Omega = [0,1]$, and let $\mathcal{F}$ be the Borel $\sigma$ field of $[0,1]$. Also let $\mathbb{P}$ denote the Lebesgue measure on $\Omega$ ($\mathbb{P}(\Omega) = 1$). Let $\mathcal{F}_n$ ...
0
votes
0answers
34 views

How to tell if a function is in $L^1$ or in $L^2$?

Given a function, what properties do I need to satisfy to tell whether a function is in $L^{1}(R)$ and/or $L^{2}(R)$ For example, $\frac{sin(x)}{|x|^{\frac{3}{2}}}$. Do I just need to test if the ...
1
vote
2answers
137 views

prove that $\int _{-\infty }^\infty \frac{1}{1+f(x)}\operatorname{dx}$ diverges

Let $f : \Bbb R \to [0,\infty )$ be a measurable function. If $\int _{-\infty }^\infty f(x)\operatorname{d}x= 1 $ then prove that $\int _{-\infty }^\infty \frac{1}{1+f(x)}\operatorname{d}x= ...
4
votes
3answers
79 views

Show that $ f (x) \to 0$ as $|x| \to \infty$.

I am learning Measure Theory .However I got stuck on follow Let $f $ be a uniformly continuous real valued function on the real line $\Bbb R.$ Assume that $f $ is integrable with respect to the ...
0
votes
0answers
26 views

Proof of $L_q(\Omega) \subset L_p(\Omega)$ when $1\leq p \leq q \leq \infty$ [duplicate]

I need to prove that $L_q(\Omega) \subset L_p(\Omega)$ when $1\leq p \leq q \leq \infty$ when $\Omega$ is bounded. The hint given is that I should use Hölders inequality. How do I start my proof ...
4
votes
1answer
158 views

A differentiation under the integral sign

Let $f:\mathbb{R}^n\to\mathbb{R}$ be a function Lebesgue summable on all $\mu$-measurable and bounded subsets of $\mathbb{R}^n$, where $\mu$ is the usual Lebesgue measure defined on $\mathbb{R}^n$, ...
1
vote
1answer
15 views

Define $Y(E,f)=\{\int_E \phi dm: 0\le \phi \le f, \phi \; \text{is simple}\}$. Show that the set $Y(E,f)$ is always an interval.

Let $f$ be a nonnegative measurable function and $E\in \mathcal{M}$, where $\mathcal{M}$ is the sigma algebra of Lebesgue measurable sets of $\mathbb{R}$. Define $Y(E,f)=\{\int_E \phi dm: 0\le \phi ...
1
vote
1answer
61 views

Prove that if $\int_E f \cdot g = 0$ then $ f=0 $

Let $E$ be a measurable set, $1\le p < \infty$, $q$ the conjugate of $p$, and $\mathcal S$ a dense subset of $L^q(E)$. show that if $f \in L^p(E)$ and $ \int_E f \cdot g = 0 \; \forall \; g \in ...
1
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0answers
24 views

Find measure of a set

I need to calculate the measure of set $A=\{(x,y,z)\in R^3: 4x^2+y^2<4,x>0,x^2>z>0\}$In other words i need to calculate integral $\int_{A}1d\lambda_{3}$.Would be more than glad for ...
1
vote
1answer
25 views

Lebesgue Iterated integral

I'm trying to evaluate this integral: $$\int_A\int_B1_{x\neq y} \, d\mu(x) \, d\mu(y)$$ where $\mu$ is the Lebesgue measure. I think the answer should be $\mu(A)\mu(B)$ but I'm not sure how to ...
2
votes
3answers
100 views

Natural ways in which the *complex* valued L-integral and *complex* Hilbert spaces come up

I have two questions regarding how two concepts that involve complex valued functions may come up in a natural way. (Non-natural ways are: These concept come up in order to present a unified theory, ...
0
votes
1answer
93 views

which of the following are true among the four statements? [closed]

let {$f_n$} be sequence of integrable functions defined on an interval $[a,b]$. Then a) If $f_n(x)\to 0$ a.e., then $\int_{a}^{b}f_n(x)dx \to 0$ b) If $\int_{a}^{b}f_n(x)dx \to 0$ then $f_n(x)\to ...
1
vote
1answer
31 views

construction of the gamma function

I have some questions regarding the construction of the gamma function below. First, why is $|t^{z-1}|=t^{\Re z-1}$? Next, why is $|f_z(t)|\le C_z e^{-t/2}$ for $t\ge 1$, and how can the precise value ...
0
votes
0answers
22 views

$f=\sum_1^\infty n^{-1}(-1)^n\chi_{(n,n+1]}$ is not Lebesgue integrable but improper Riemann integrable

Show that $f=\sum_1^\infty n^{-1}(-1)^n\chi_{(n,n+1]}$ is not Lebesgue integrable but $\lim_{b\to \infty} \int_0^b f(x) dx$ (the limit of the Riemann integral) exists. I'm stuck on how to show this. ...
2
votes
0answers
77 views

The integral $\int f \mu(dt)$ versus $\int f \mu(t) dt$

In the theory of continuous-time branching processes there is a quantity called the Malthusian parameter $\alpha \in \mathbb{R}$ defined by: $$ \int_0^{\infty} e^{-\alpha t} \mu(dt) = 1 $$ The ...
1
vote
1answer
32 views

$0\le f_n\le g_n\le h_n$ $f_n$ and $h_n$ are Lebesgue integrable then $g_n$ is also.

I was trying to solve the following problem but unfortunately I'm unable to solve it. Please help me. Let $f_n,g_n,h_n$ be Lebesgue integrable functions with $0\le f_n\le g_n\le h_n$ for all $n$. ...
0
votes
0answers
38 views

Approximation of Integrable function

Let $f$ be an integrable function over $E$. Then given $ε > 0$, there is a simple function $φ$, step function $ψ$, and a continuous function $g$ supported on a finite measure set such that 􏰁 􏰁 􏰁 ...
1
vote
1answer
37 views

Compare measures of two sets

I am struggling with probably something very trivial. Consider a dynamical system whose evolution equation is given by: $x^+ = f(x)$, where $x,x^+\in R^n$ and $f(x)\in R^n$ is piecewise continuous ...
1
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1answer
52 views

Monotone convergence theorem (measure theory) proof

I am stuck with the final line on the image, why does $f_n \to f$ (which also increases), mean that $\cup_n A_n = \Omega$?
0
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0answers
10 views

Approximation via step functions through the conditional expectation on a sigma-field?

Suppose that $B$ is a Borel sigma field on $[0,1]$ and that we are working with Lebesgue measure. Also, suppose that $B_n$ is the smallest sigma field containing the intervals $I_j = [j2^{-n}, ...
3
votes
0answers
12 views

$q$-integral of product of two functions

How to solve $\int\limits_a^b \left[\int\limits_t^b (x-tq)_{\Re(\alpha n+\beta-1)} d_qx\right]|\phi(t)|d_qt$ and what will be the answer? Where $|q|<1$ and the function $\phi$ be in the space ...
0
votes
0answers
28 views

Is an $o\left(\frac{1}{x}\right)$ for $x \to 0$ function Lebesgue integrable in a neighbourhood of $0$?

Let $F:(-1,1)\setminus \{0\} \to \mathbb R$ be a continuous function such that $\lim_{x\to 0} F(x) = + \infty$ and $\lim_{x\to 0} x F(x) = 0$. Can we say that $F \in L^1(-1,1)$ ?
0
votes
0answers
49 views

Is $\frac{1}{\sqrt x}$ locally integrable?

Can we say since its anti-derivative is like $\frac{1}{2}\sqrt x$, even if we study the measurable bounded set near zero, the anti-derivative will not go to infinity, and hence the original function ...
1
vote
1answer
58 views

Almost everywhere continuous functions

A function that is almost everywhere continuous is in $L_2$; however, the converse might not be true. I couldn't find any example to show this, could you help me with this? $$L_2= \left \{ ...
0
votes
1answer
25 views

Calculating surface integral (with gauss's theorem (?))

I would like to solve the following problem: Let $B_1$ be the unit ball in $R^3$ and $A := \delta B_1 \cap(\{x>0, z=0\}\cup\{x=0, z>0\})$. Let $F(x,y,z) := (-y+e^{x+z}, 0, e^{x+z})$. ...
1
vote
1answer
39 views

Are these functions Lebesgue Integrable over these domains?

link to original image $\frac{\sin x^2}{x}$ on $(1,\infty)$; $\frac1x\sin\frac{1}{x^2}$ on $(0,1)$. Hi, I've managed to prove that $\frac{\sin(x)}{x}$ is integrable on $(0,R]$ and is not ...
0
votes
2answers
32 views

Calculating 2-dimensional integral

I would like to calculate $\int_A (6yx+2y) d(x,y)$, where $A:= \{(x,y) \in R^2 | x^2/4 + y^2 \leq 1\}$. Can anybody help me with this problem?
0
votes
2answers
69 views

Show these functions are/ aren't Lebesgue Integrable? How do I go about showing this?

I have recently been learning the comparison test, MCT, Fatou's Lemma and DCT for Lebesgue integrals, but have been struggling with the details of the proofs. 1) f is 0 a.e. so is integrable to 0 ...
3
votes
3answers
65 views

If $f$ is bounded non-negative $L^1$, is $f\leq g$ a.e. for some continuous integrable $g$?

Suppose $f\in L^1(\mathbb{R})\cap L^\infty(\mathbb{R})$ is bounded, non-negative and integrable (w.r.t. Lebesgue measure) : does there exist $g$ continuous (non-negative) and integrable such that $$ ...
1
vote
1answer
42 views

If a sequence of functions is zero almost everywhere and converges pointwise almost everywhere, does the same hold for a limit?

Let $(f_n)_{n \in \mathbb{N}}$ be a sequence of functions in $\mathcal{L}^p(\mathbb{R})$. Each $f_n$ is zero almost everywhere. Additionally, the sequence converges pointwise almost everywhere to some ...
1
vote
1answer
35 views

Is the subspace of $L^2([0,1])$ of all “functions” vanishing on $[0,1/2]$ closed?

I am trying to understand the following example from my lecture notes: $\mathcal{H} := L^2([0,1],\lambda)$, then $$K := \{f \in \mathcal{H} \colon f(x) = 0, \text{ for } 0 \leq x \leq ...
2
votes
1answer
11 views

Necessity of measurable property

Consider the definition of the Lebesgue integral for a positive function $X\rightarrow [0,+\infty]$: $$ \int f(x) d\mu=\sup_{g\in S, \forall x : g(x)\leq f(x)} \left(\int g(x) d\mu \right)$$ where ...
1
vote
1answer
86 views

Proof of regular version of the Urysohn lemma

I know it's a well-known result, but I have not found any clear formalization, and I need a clear formalization. So I want to know if you agree with this formalization, and this proof. Thank you for ...
0
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0answers
31 views

Theorem 5.3 in Royden 3rd real analysis, pages 100 — 101.

The theorem stats that Let $f$ be an increasing real-valued function on the interval $[a,b]$. Then $f$ is differentiable almost everywhere. The derivative $f'$ is measurable, and $\int^{b}_a f'(x) ...
0
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0answers
18 views

Lower bound of a Lebesgue measure set

Given a sequence of times $T_n\rightarrow \infty$ and $\int_{-T_n}^{T_n} ||u_n(t)||_{L^\infty(R)} dt\geq cT_n$ for some constant $c$. Also, assume that $u(t,x) = O(1)$ for all $t,x\in R$. Prove that ...
2
votes
1answer
42 views

prove absolute integrability given square integrability

am trying to follow the outline of a proof in a book i am reading - must be missing something obvious, but would like to understand what exactly... $f$ is complex and square integrable over e. g. [0, ...
0
votes
1answer
47 views

Prove that $\int_E fd\mu = \lim \int_E f_n d\mu$ for all measurable set $E$

This is problem 4T in Bartle's The elements of integration and Lebesgue measure. Suppose $f_n$ are non-negative measurable function such that $(f_n)$ converges to $f$, and that $$\int fd\mu = ...
0
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0answers
15 views

Square Integrable Functions Formula

The book reads as follows: "Let $a(x)$ and $b(x)$ be square integrable functions defined on [$a , b$]. First we note that it follows from the elementary inequality $|ab| \le 1/2 (a^2 + b^2)$ ...
1
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2answers
49 views

Integral of a measurable function

I do not know what should i keep as title for this question... Question goes like this.. Let $f:\mathbb{R}\rightarrow [0,\infty)$ be a measurable function. If $\int_{-\infty}^{\infty}f(x)dx=1$ prove ...
0
votes
2answers
53 views

In which way Lebesgue Integral integrates over values?

A special tutorial for full dummies says In order to distinguish between the Lebesgue and Reimann integrals consider the values that the function f can take to be on the x-axis (called the ...
2
votes
1answer
31 views

Measurablity of functions defined over sections of product measures

I have to solve the following exercise but I am unable to proceed. Could you please give me some hints to how to solve it? Let $(\Omega_1, \mathcal{F}_1)$ and $(\Omega_2, \mathcal{F}_2)$ be ...
0
votes
1answer
36 views

Prove the following integral is asymptotically zero

I have to solve the following exercise. I would appreciate to get a hint for it. Suppose $(\Omega, \mathcal{F}, \mu)$ be a measure space. Let $f$ be an integrable function. Show ...
0
votes
1answer
39 views

Local Riesz Potential estimate in terms of Maximal Function

For $f \in L^1_{\text{loc}}(\mathbb R^n)$, and fixed $R > 0$ we defined the local Riesz potential by $$I(x) = \int_{B(x,R)} \frac{f(y)}{\lvert x-y \rvert^{n-1}} d\lambda (y), \hspace{1cm} x \in ...
1
vote
1answer
40 views

Continuity of $F(x)=\int_{(-\infty,x]}fd\lambda$

For a homework assignment I was told to prove that given $f\in L^1(\mathbb R)$, the following function is continuous $$F(x)=\int_{(-\infty,x]}fd\lambda.$$ I thought to use DCT and show sequential ...
2
votes
2answers
61 views

Why is Monotone Convergence Theorem restricted to a nonnegative function sequence?

Monotone Convergence Theorem for general measure: Let $(X,\Sigma,\mu)$ be a measure space. Let $f_1, f_2, ...$ be a pointwise non-decreasing sequence of $[0, \infty]$-valued ...
0
votes
0answers
45 views

Show $\frac{y-x}{(2-x-y)^3}$ is not integrable on $[0,1]\times[0,1]$, not invoking Fubini's theorem.

The double integral $$I = \int_{[0,1]\times[0,1]}\frac{y-x}{(2-x-y)^3} dxdy$$ does not have a finite value. The two iterated integrals have different values (Counterexample to Fubini?). Then Fubini's ...
4
votes
0answers
61 views

Lebesgue versus Riemann integrable

Can a Lebesgue measurable function be modified on a set of first category so as become continuous except on a set of Lebesgue measure zero? OR Can a Baire-measurable function be modified on a set of ...
1
vote
2answers
36 views

Could fast or irregular oscillations make Lebesgue integral fail?

Let's consider real measurable functions defined in a bounded interval. As long as a function is bounded, oscillations at least cannot make the volume under the graph of the function infinite. But I'm ...
1
vote
2answers
42 views

$f$ has a zero integral on every measurable set. Prove $f$ is zero almost everywhere

I am trying to solve the following exercise: Let $f$ be integrable. Assume that $\int_A f d\mu = 0$ for every measurable set $A$. Prove that $f = 0$ a.e. [$\mu$]. I have the following proof but it ...