Tagged Questions
1
vote
0answers
29 views
Show derivative of integral equals integral of partial derivative if M[0,1]-measurable
I am trying to determine a method of approaching the following:
Suppose that $f:[0,1] \times (0,1)$ $\rightarrow$ $\mathbb{R}$ is such that, for each $y \in (0,1)$, the function $f^{[y]}(x) = f(x,y)$ ...
1
vote
0answers
34 views
question about integral over compact set and bound
Suppose $f(t)$ exists and is finite for a.e. $t \in [0,T]$. Then can I say that $\int_0^T f(t)h(t) \leq K\int_0^T |h(t)|$ for some constant $K$? I know nothing about whether $f$ is continuous, but ...
0
votes
3answers
33 views
Dirichlet function involved with integral
Let the Dirichlet function $D:[0,\pi] \rightarrow R$ be given by
$D(x):=\begin{cases} 0 &\text{if } x\in [0,\pi] \cap Q, \\{}\\ 1 &\text{ otherwise}.\end{cases}$
For the function ...
2
votes
1answer
42 views
How to recover a measure from its Fourier transform?
Let $f$ be the complex function defined on $\mathbb{R}$ by
$$
f(t)=\frac{1-it}{1+it}.
$$
1) Does there exist a complex bounded measure $\mu \in M(\mathbb{R})$ such that $\hat{\mu}=f$ (where $\hat{}$ ...
3
votes
0answers
76 views
Positive functions with zero integrals
I was a bit confused by this link mentioned in this question - in particular, in Remark 4.21:
Suppose that $f$ is a positive function on $[a,b]$. If $f$ is Henstock-Kurzweil integrable, then the ...
3
votes
1answer
84 views
Show $g(\mathbf{x}) \leq h(\mathbf{x})$ implies $\int g(\mathbf{x})\mathrm{d}\mathbf{x} \leq \int h(\mathbf{x})\mathrm{d}\mathbf{x}$
Suppose I have $g$ and $h$ from $\mathbb{R}^p\to\mathbb{R}$ such that for all $\mathbf{x}$, $g(\mathbf{x}) \leq h(\mathbf{x})$. I want to prove that the integral over all $\mathbb{R}^p$ of $g$ is less ...
0
votes
1answer
38 views
Question regarding application of Tonelli Theorem
Hi I have a question below, I am wondering if anyone would help me with it, thank you in advance!
Prove that for any independent random variable x, y then
$$\int_{\mathbb{R}}F_{x} dP_{y} = ...
-2
votes
2answers
63 views
Combinations of integrable functions
If $f$ and $g$ are integrable functions and real-valued on $(X,M,\mu)$ , which assertion is correct?
$fg\in L^1 (\mu)$
$fg\in L^2 (\mu)$
$\sqrt{f^2 +g^2}\in L^1 (\mu)$
None of ...
1
vote
0answers
42 views
Using $\frac{1}{A+i\epsilon} = PV\frac{1}{A}-i\pi\delta(A)$ in Feynman Integrals
Is the following operations OK (this is related to the Feynman parameter trick)?
$$F:= \int_0^1 \mathrm{d}x\int_0^{1-x}\mathrm{d}y \frac{1}{f(x,y)+\mathrm{i}\epsilon}.$$ Now using
...
6
votes
4answers
287 views
Meaning of measure zero
My book describes measure zero as following:
A set of points on the $x$-axis is said to have measure zero if the sum of the lengths of intervals enclosing all the points can be made arbitrarily ...
0
votes
1answer
51 views
A question about integral and equivalent measures
Let $\lambda$ be the Lebesgue measure and $\mu$ be a probability $\sigma$-finite measure on $[0, 1].$ Suppose $\lambda \ll \mu$ and $\mu \ll \lambda.$ What can we say about the convergence of the ...
0
votes
1answer
54 views
A question about Integral and Measure theory
Lets consider the integral $\int_{0}^{1}logxd\mu$ where measure $\mu$ is equivalent to the Lebesgue measure. What about convergence of this integral? Thanks.
2
votes
1answer
124 views
Characteristic function of the Smith-Volterra-Cantor set
Let the characteristic function of the SVC set be denoted by $ \beta $. Does the Riemann integral $ \displaystyle \int_{0}^{1} \beta ~ d{x} $ exist? I think it does since $ \beta $ is bounded, but I ...
0
votes
1answer
56 views
The relation between arbitrary measure space and the Lebesgue integral
Let $(X, \mathcal F, \mu)$ be a measure space and $f\in M^+(X,\mu)$ (the measurable non-negative functions), and $t>0$. Now let $$S_f(t)=\{x\in X:f(x)>t\} \quad \Psi_f(t)=\mu(S_f(t))$$
Prove ...
4
votes
0answers
145 views
“Simple” proof of Lebesgue's Differentiation Theorem for dimension 1?
Lebesgue Differentiation Theorem for $\mathbb{R}$: Let $f:[a,b]\to \mathbb{R}$ be intergable and $F(x)=\int_a^xf$. Then $F$ is differentiable almost everywhere in $[a,b]$ and $F'=f$ a.e.
Is there a ...
1
vote
0answers
51 views
Infinite function on measure zero set
I am trying to prove that given a set $E$ of measure 0, and a function $f \equiv \infty$ on $E$, then $\displaystyle \int_{E}f = 0$.
This would be easy if one is allowed to assume that $\infty \times ...
2
votes
1answer
86 views
Bounded measurable function and integral with charcteristic function
I have been struggling with the following for quite some time now. If anyone can give me some help, it will be much appreciated:
Let $f$ bounded, measurable and $E$ be a set of finite measure. Let $A ...
1
vote
1answer
126 views
Prove that the indicator function for $\mathbb{Q}\cap[0,1]$ is not Riemann integrable
Prove that the function $ \phi_\mathbb Q : [0,1] \to \{0,1\}$ defined by
$$\phi_\mathbb{Q}(w)=\begin{cases}
1 &\text{if } w\in \mathbb Q, \\
0 & \text{if } w\notin \mathbb ...
0
votes
1answer
93 views
show that a sequence of functions is bounded by an integrable function
show that the sequence-indexed with $a_n$ , $${1\over{1+t^2}} - {e^{-ta_n}\over{(1+t^2)}}(\cos a_n + t\sin a_n)$$ is bounded from above by an integrable function for a sufficiently large $a_n$
...
3
votes
0answers
72 views
Integration of sine^2 w.r.t. some norm
Let $||x||$ be any norm over $\mathbb R^n$. Let $B_T$ the open ball with radius $T$ w.r.t. to our norm, i.e. all $x\in\mathbb R^n$ such that $||x||<T$. Let $n\in\mathbb N$.
How much ...
0
votes
0answers
49 views
How to determine point of zero measure?
Today in our physics lecture, our Prof told us during some calculation that for $x\rightarrow0$
$f(x)\rightarrow\frac{1}{x^2}$
which was easily understandable from the context and our previous ...
2
votes
0answers
124 views
definition of operator valued integral with spectral measure
I am trying to make sense of some operators that come up on Buchholz and Summers' work on warped convolutions (two works on arxiv: 2008 and 2011).
There, they work on a Hilbert space $H$ and on the ...
1
vote
0answers
22 views
Construction of a finite real-valued Lebesgue measurable function with infinite integral on every open set? [duplicate]
Possible Duplicate:
examples of measurable functions on $\mathbb{R}$
I am trying to construct a Lebesgue measurable function $f: \mathbb{R}\rightarrow\mathbb{R}$ which has the property that ...
1
vote
0answers
66 views
Lebesgue measure is invariant under isometry
Is it true that Lebesgue measure is invariant under isometric map? I mean standard measure of $R^n$.
It is certainly true for interval in $R$ (obvious). I've attempted to prove it in general by ...
0
votes
0answers
37 views
condition for limit and differential
let $f:X\times [a,b] \rightarrow \mathbb R$. find conditions under which you may assert each of the following:
$$
\lim_{t \to t_0} \int_X f(x,t)d\mu = \int_X \lim_{t \to t_0} f(x,t)d\mu
$$
$$
{d ...
3
votes
1answer
61 views
Uniform integrablity of measurable functions
How can I show that if family of $f$ is uniformly integrable then so is {$|f|$}?
$($by uniformly integrablity: $\forall \epsilon>0 \ \exists \delta>0: |\int_Ef|<\epsilon,\mu(E)<\delta)$
...
4
votes
1answer
166 views
Extension of Fatou's lemma
let $X$ be a finite measure space and $\{f_n\}$ be a sequence of integrable functions, $f_n \rightarrow f\text{ a.e.}$ on $ X$.
I want to show if (1) holds, then (2) holds too.
$$\lim_{n \rightarrow ...
3
votes
1answer
97 views
Does $\lim_{n\rightarrow \infty} \int_X f_n - \int_X f\gt 0$ implies that convergence of $f_n$ to $f$ a.e. fails?
I've come across this problem as a part of another proof that I'm writing and I want to know if this is a right conclusion:
Let $X$ be a finite measure space and $\{f_n\}$ be a sequence of ...
5
votes
1answer
126 views
Compute $\lim_{n\to\infty}\int_0^n \left(1+\frac{x}{2n}\right)^ne^{-x}\,dx$.
I'm trying to teach myself some analysis (I'm currently studying algebra), and I'm a bit stuck on this question. It's strange because of the $n$ appearing as a limit of integration; I want to apply ...
4
votes
1answer
48 views
Inequality between Probability and Expectation
I have to prove an inequality between probability and expectation and I wanted to ask for help on it. Here is the problem:
Assume that $Y \ge 0$ and $E Y^2 < \infty$. I need to prove that:
...
1
vote
1answer
74 views
Stochastic integral: $E\left(\int^1_0(W(s))\,ds\int^1_0t(W(t)\right)\,dt$
I need to calculate the expectation of the product between the integral of a Wiener process and the expectation of a Wiener process. Is the same as the expectation of the product between the integral ...
1
vote
0answers
45 views
A measure realated to Riemann integral
Let $( \mathbb{R}^k , \mathcal{A} , m_{k} )$ be a Lebesgue measurable space, i.e., $m_{k}=m$ is a Lebesgue measure. Let $f: \mathbb{R^k} \to \mathbb{R}$ be a $m$-integrable function. Define a function ...
1
vote
2answers
73 views
Prove $ \int_{cX} \frac{dt}{t} = \int_{X} \frac{dt}{t}$ for every Lebesgue measurable set $X$
Let $c>0$. Let $X \subseteq (0,\infty)$ be a Lebesgue measurable set. Define $$ cX := \{ cx \mid x \in X \}. $$ Then $$ \int_{cX} \frac{dt}{t} = \int_{X} \frac{dt}{t}$$
Now I can prove this for ...
7
votes
5answers
193 views
Proving an estimate for this integral
How can I show that$$\sqrt[3]6>\int_1^\infty\frac{(1+x)^{1/3}}{x^2}\mathrm dx?$$
3
votes
0answers
151 views
Generalized Change of Variables Theorem?
Is there a generalized form of the differentiable change of variables theorem for Lebesgue integrals? That is, if we consider the well known change of variables theorem: If $\phi : X \rightarrow X$ is ...
3
votes
0answers
121 views
Equivalence of integrals
Let $x_1, \ldots, x_n$ be vectors in the normed space $(X, \|\cdot\|)$.
Let $\mu$ be the Lebesgue measure on the cube $[-1,1]^n$. Denote vectors in $[-1,1]^n$ by $y=(y_1, \ldots, y_n)$.
Are the ...
0
votes
1answer
112 views
Seeking clarification of Lebesgue definition given for $\int _{0}^{1}x^{-a}dx$
I came across the example
"Show that $\int _{0}^{1}x^{-a}dx$ exists as a Lebesgue integral, and is equal to $1/(1-a)$, if $0 < a < 1$; but is infinite if $a\geq 1$.
The Lebesgue definition of ...
2
votes
1answer
74 views
When does non-negativity of the integral of a function imply that the function itself is non-negative?
Let $(\Omega,\Sigma)$ be a measurable space and $(\omega_k)_{k\in\mathbb{N}}$ a sequence of elements of $\Omega$. Let
$$
\mathcal{M}:=\left\{\sum_{k=1}^\infty a_k\cdot\delta_{\omega_k}: ...
0
votes
1answer
69 views
Convergence of Lebesgue integral with negative part finite
Let $(f_{n})_{n}$ a sequence in $\mathcal{L}^1(\mathbb{R})$ and $f_{n}<f_{n+1}$. Also $\int_{\mathbb{R}}f_{k}^-dm < \infty$ for some $k\in \mathbb{N}$. Show that ...
3
votes
2answers
148 views
Proof of $\int\limits_{A}f=\int\limits_{\mathbb{R}}f{1}_A$ for the Lebesgue integral
Let $f:\mathbb{R}\to [0,\infty]$ be a measurable function and $A\subset \mathbb{R}$. Then, show that
\begin{equation}
\int\limits_{A}f=\int\limits_{\mathbb{R}}f{1}_A \tag{1}
\end{equation}
where ...
2
votes
2answers
115 views
Integral variable substitution using Hausdorff measure
Suppose we have positive density $q$ with "good" qualities (continuity, etc..). I need to calculate this integral:
$$\int_B q(\textbf{z}) d \textbf{z},\ \textbf{z} \in \mathbb{R}^d,$$
where $B \subset ...
0
votes
0answers
54 views
The coarea formula for topological groups
The classical coarea formula provides us a possibility to reduce the integration over some set to the integration over the slices of this set. For example, we can reduce the integration over a unit ...
2
votes
2answers
158 views
Integral from ball to sphere
Well I am working something, which deals with the following problem: For example, I want to compute an integral $\int_{B(0,R)}f(x)dx$, where $B(0,R)=\{x\in\mathbb R^n:\;|x|\leq R\}$ and ...
2
votes
0answers
223 views
Lebesgue Line Integrals - Parametric Change of Variables
Consider the following Lebesgue integral in $\mathbb{R}^n$
$$ \int_C f(x) dx $$
Where $f : \mathbb{R}^n \rightarrow \mathbb{R}$ is measurable and $C$ is a measurable subset of $\mathbb{R}^n$ that ...
6
votes
1answer
388 views
Summing over General Functions of Primes and an Application to Prime $\zeta$ Function
Along the lines of thought given here, is it in general possible to substitute a summation over a function $f$ of primes like the following:
$$
\sum_{p\le x}f(p)=\int_2^x f(t) d(\pi(t))\tag{1}
$$
and ...
1
vote
1answer
84 views
Equality of measures
I have two measures $\mu$ and $\nu$ supported on compacts in $\mbox{int } \mathbb{R}^{n}_+$. Are there some sufficiently general classes of such measures for which
$$
\int\limits_{\mathbb{R}^n_+} ...
9
votes
3answers
547 views
What is the insight behind the Lebesgue integral?
Edit 3: OK, I had an insight, inspired in part by Ben-Blum Smith's comment, and the post he linked to. (I have no idea if this insight is right; it's barely a hunch, and that's why I'm not submitting ...
4
votes
1answer
95 views
Is $f$ non-decreasing a.e. if its primitive is convex?
The subsequent statement can be regarded as a follow-up to
If $\int_0^x f \ dm$ is zero everywhere then $f$ is zero almost everywhere
Is $f$ non-negative a.e. if its primitive is non-decreasing?
...
2
votes
2answers
158 views
Is $f$ non-negative a.e. if its primitive is non-decreasing?
Let $f:[a,b]\to\mathbb{R}$ be Lebesgue integrable.
Clearly, if $f$ is non-negative then
$$
g:[a,b]\ni x\mapsto\int_a^x f(t)\,\mathrm{d}t\in\mathbb{R}
$$
is non-decreasing since for $x<y$ it ...
2
votes
1answer
101 views
Convergence of Lebesgue integrals involving trigonometric functions.
For $m \in \{1,2,\ldots\}$, I have the sets $A_m=\{x \in [0,1]^2: |x|^2 \geq \frac{1}{m^2},x_1^2 \leq x_2 \leq \sqrt{x_1}\}$, $A_\infty=\{x \in [0,1]^2: x_1^2 \leq x_2 \leq \sqrt{x_1}\}$ (I use the ...
