# Tagged Questions

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### Positive integral everywhere implies positive function a.e

I would like to get feedback on my demonstration of this simple statement : Let $f$ be an integrable function on the measure space $(X,S,\mu)$. \begin{align} \text{If }\int_E f \, d\mu \geq 0\text{ ...
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### Triangle inequality for integrals with complex valued integrand

This is a step in a lecture note I'm reading. It should be simple because the author considers it obvious but I can't see it. What am I missing? Suppose $U$ and $V$ are integrable over measure space ...
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### Hardy Littlewood maximal function and integral comparison.

Define the Hardy Littlewood maximal function $$g^*(y)=\sup \left\{\frac{1}{|B|}\int_B|g(x)|dx:B\text{ is any open ball containing y}\right\}.$$ For given $x_i,r_i,a_i$, first I have shown that ...
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### Inequality with moments

Let $m$ a probability measure, $f$ a positive measurable function (one can assume it is bounded, the existence of the moments is not a problem here). Is $m(f^3) \le m(f^2) m(f)$?
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### How to apply the Hölder's inequality in a clever way?

Here is the problem: Let $f\in L^p(\mathbb R^n)\cap L^q(\mathbb R^n)$ and $s\in[p,q]$. Show that $f\in L^s(\mathbb R^n)$ I'm almost sure that this is a simple exercise on Hölder's inequality yet ...
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### Caratheodory's theorem and outer measure

I'm trying to show that $$\lambda(A)=\lambda(A\cap E)+\lambda(A\cap E^c)$$ where $\lambda$ is an outer measure, $A\subset \mathbb{R}$, $E \subset \mathbb{R}$, and $E$ is an elementary set; that is, ...
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### Generalized Hölder inequality, the case when equality holds

I know the generalized Hölder inequality sounds like as: Let $1\leq p_1,\ldots,p_n<\infty$ and $p>0$ such that $\frac1p=\frac1{p_1}+\cdots+\frac1{p_n}$. Then, for all measurable functions ...
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### Find $\inf_{f > 0} T_f := \left(\int_A f \, d\mu\right)\left(\int_A \frac{1}{f} \, d\mu\right)$

This exercise gives me trouble: Let $F$ denote the collection of measurable functions which are positive $\mu$-a.e. and let $A \in \mathbb X$ satisfy $0 < \mu(A) < \infty$. For $f \in F$ let ...
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### Absolute value bound of Lebesgue integral

For the Riemann integral, we have the bound $$\left|\int_Af(x)dx\right|\leq\left(\sup_{x\in A}|f(x)|\right)\cdot\left|\int_Adx\right|$$ Do we have a similar bound for the Lebesgue integral, one like ...
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### Proof of the nonexistence of an identity $\phi$ involving convolution

The Banach space $L^1(\mathbb{R}^n)$ is an algebra with a product (convolution) which is both commutative and associative. But this algebra does not have a multiplicative identity. An attempt to show ...
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### Interesting inequality $\|F\|_p\le \frac{\pi}{\sin(\pi/p)}\|f\|_p$ over $L^p$

Consider the function $$F(x)=\int_0^\infty \frac{f(y)}{x+y} \, dy, \quad0<x<\infty$$ Prove that if $1<p<\infty$, $$\|F\|_p\le \frac{\pi}{\sin(\pi/p)}\|f\|_p$$ and show that the constant ...
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### Intuition behind Gaussian isoperimetric inequality

I was wondering whether or not there's an intuitive way of understanding the Gaussian isoperimetric inequality. I have been studying the Classical isoperimetric inequality and I finally understand it. ...
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### Proving an estimate for this integral

How can I show that$$\sqrt[3]6>\int_1^\infty\frac{(1+x)^{1/3}}{x^2}\mathrm dx?$$
How can I prove the following inequality: (where $f$ is nice enough) - Given a function $f(x,y) : \Omega_1 \times \Omega_2 \to \mathbb{R}$ , and $\alpha,C_1,C_2$ are some constants, ( $\Omega_i$ ...