# Tagged Questions

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### 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 $f^{[y]}(x) = f(x,y)$ ...
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### 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 ...
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### 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 ...
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### 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 $\hat{}$ ...
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### 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 ...
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### 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 ...
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### 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 $a_n$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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: ...
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### 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 ...
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### 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 ...
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### 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 ...
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### Proving an estimate for this integral

How can I show that$$\sqrt[3]6>\int_1^\infty\frac{(1+x)^{1/3}}{x^2}\mathrm dx?$$
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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? ...
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### 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 ...
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### 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 ...