Continuity and lebesgue integrability of integral function, proof verification
Suppose that $f\in L^1(\mathbb R, \mathcal B_{\mathbb R} ,m)$ where $m$ is the standard lebesgue measure. For fixed $h$, let us define:$$\phi(x)= \frac{1}{2h} \int_{x-h}^{x+h} f(t) dt $$Show that...
View ArticleWhat is this set? Unknown baire generic set
I'm interested in the set given in this question. For clarity, set$$S_{\epsilon} = \cup_{j = 1}^{\infty} \left( q_j + \frac{\epsilon}{2^j}, q_j - \frac{\epsilon}{2^j}\right)$$where $\\{q_j\\}$ is an...
View ArticleMissing argument in the proof of the Levy-Khintchine representation .
In one proof of the Levy Khintchine representation of the Laplace exponent of subordinators, the following argument is used:"Assume $f_n : [0,\infty) \to [0,\infty)$ is a sequence of non-increasing...
View ArticleProperties of sup and inf for bounded sequences [closed]
Let $(a_n)$ and $(b_n)$ be bounded sequences. How to prove that:(a) $\sup_{k \ge n}(a_k+b_k) \le\sup_{k \ge n}a_k +\sup_{k \ge n}b_k$.(b) $\inf_{k \ge n}(a_k+b_k) \ge\inf_{k \ge n}a_k +\inf_{k \ge...
View ArticleCauchy-Schwarz in a complex Hilbert Space proof question
my instructor gave me the following proof for the Cauchy-Schwarz inequality in a complex Hilbert Space, but I do not understand the following part:With a fixed value of $|\lambda|$, the left side is...
View ArticleContinuity of the right shift
In the lecture we consider the right shift$$u:\mathbb{R}\rightarrow X$$$$ t \mapsto f(\cdot -t)$$ for fixed $f \in X$. If we take for $X = (C^0_b,\Vert \cdot \Vert_\infty)$, hence the space of...
View ArticleHow do I Show that $\exists a, \forall e > 0; a < e \Rightarrow a \le 0$
When proving using contradiction,can the method specified below be used.lets assume that $\forall a, \exists e > 0; a < e$ and $a > 0$since the condition holds for all 'a', let $a=2e$then...
View ArticleThe radius of convergence using root test
Is the root test sufficient for finding the radius of convergence of any power series ,i.e, does it determine the whole area where a power series converges? if the answer is no can you give an example...
View ArticleHow to get the following estimate of integral invoving Airy function
$$\mbox{Define}\quadG(x,y)=\frac{Ci\left(\gamma(x-y_c)\right)Ai\left(\gamma(y-y_c)\right)}{\epsilon\gamma},$$where $y>x,$$y_c$ is a complex number such that $\Re(y_c)>0,$$\epsilon$ is a pure...
View ArticleInfinite Summation of Almost Sure Convergent RVs
Suppose we have random variables $X_{n,i}$ for $n\geq1,i=1,\dots,b_n$. Here we suppose $b_n$ is non-decreasing. We know that for each $i$, the sequence $X_{n,i}$ converges to $X_i$ almost sure.Now we...
View ArticleHow to give this sum a bound?
Let $x,y\in\mathbb{Z},$ consider the sum below$$\sum_{x,y\in\mathbb{Z}\\ x\neq y}\frac{1}{|x-y|^{4}}\Bigg|\frac{1}{|x|^{4}}-\frac{1}{|y|^{4}}\Bigg|,$$is there anything I could do to give this sum a...
View ArticleCompletely monotone function is analytic
I want to prove the following.Let $I:=[a,b)$ be finite interval, $f$ is completely monotone on $I$.Then it can be extended analytically into the complex z-plane($z=x+iy$), and the function $f(z)$ will...
View ArticleApproximation of a class of measurable functions by simple functions with...
It is well known that if $f:\mathbb{R}\rightarrow \mathbb{R}$ is a measurable function and $f \ge 0$ then there exist a sequence of simple non-negative measurable functions {$ S_n $} such that...
View ArticleAnalytic sets have the Baire property
This problem is from the text book Bruckner, A. W., et. al., Real Analysis (2nd edition), page 459.A set in $A$ a topological space $X$ has the Baire property if $A=G\triangle P$ where $G$ is open, and...
View ArticleHow to show that the limit of a sequence is not equal to some value?
In the second chapter of the book Understanding Analysis by Abbott, Example 2.2.6 proved that picking $N>\frac{1}{\varepsilon}$ suffices to prove the convergence of $\frac{n+1}{n}$ to the number...
View Articleweak convergence and pointwise implies $L_p$ convergence
Suppose $f_i \to f$ weakly in $L^p(X, M, \mu)$, $1 < p < \infty$, and that $f_i \to f$ pointwise $\mu$-a.e. Prove that $f_i^+ \to f^+$ and $f_i^- \to f^-$ weakly in $L^p$.My proof:Since $f^\pm =...
View ArticleBounded linear functionals on $L^p(\mathbb{R})$, $0
Previously asked on this site: for $p\in(0,1)$, there are no bounded linear functionals on $L^p(\mathbb{R})$. I want to follow-up about what happens if we consider a general measure $\mu$; I do not see...
View ArticleTranslates of a set of positive Lebesgue measure cover $\mathbb{R}$?
Let $E$ be a set of positive Lebesgue measure in $\mathbb{R}$. Does some countable union of translates of $E$ cover $\mathbb{R}$?My intuition is that $\mathbb{R}$ can be covered with countable...
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Rudin's proof of Open Mapping Theorem (5.9 RCA, p. 100)
This is a very minor question about Rudin's proof of the Open Mapping Theorem. I have included his proof if you don't happen to have the book handy.Just before display (4), Rudin fixes $y \in \delta V$...
View ArticleIs there a sequence of real numbers so that $\sum_{n=1}^{\infty}x_n^p$...
Is there a real sequence $(x_n)$ such that the series $$\sum_{n=1}^{\infty}x_n^p$$ convergent if and only if $p$ is prime ?If the answer is YES, can we find an explicit formula for $x_n$ ?What happens...
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