$L_p$ function and improper integral
I was studying at $L_p$ spaces, and I came across this thread (among others) : Show that the following function $f \in L_p$ if and only if $p=2$Now, my question is : Why they analyze if the function is...
View ArticleExamine the convergence of $\int_0^1x^{n-1}\log x dx.$
Examine the convergence of $\int_0^1x^{n-1}\log x dx.$The solution given is as follows:$0$ is the only point of infinite discontinuity of the integrand. Let us examine the convergence of $\int_0^{\frac...
View ArticleRudin's Definition 1.7 and Theorem 1.11
Definition 1.7 in Baby Rudin says: Suppose S is an ordered set, and E $\subset$ S. If there exists a $\beta \in$ S such that $x \leq \beta$ for every $x \in E$, we say E is bounded above and call...
View ArticleDoes the existence of the iterated integral imply measurability?
Let $(X, \mathcal{M}, \mu)$ be a measure space and denote the Lebesgue measurable subsets of $\mathbb{R}$ by $\mathcal{L}$. Assume $f : X \times \mathbb{R} \rightarrow \mathbb{R}$ has the following...
View ArticleProve that $\frac{2(x^2+1)^2}{e^{x^3/3 + x}}$ is bounded and $ \int_0^\infty...
Prove that $\frac{2(x^2+1)^2}{e^{x^3/3 + x}}$ is bounded and $ \int_0^\infty \frac{2(x^2+1)^2}{e^{x^3/3 + x}} \ dx $ converges. Wolfram alpha returns me $3.44857$ for the integral, but how to prove it...
View ArticleOn the absolute continuity and the Riemann integrability of a particular...
We consider $f\in L^1[a,b]$, where $F$ is a finite interval of $\mathbb{R}$. The function $F\colon [a,b]\to\mathbb{R}$ defined as $$F(x):=\int_{[a,x]}f\;d\lambda\quad(x\in [a,b])$$it is called integral...
View ArticleIntegrals on Cantor sets
$P$ is a Cantor set. Let$$f(x)=\begin{cases}\ln(1+x^3) ~~~x\in P\\x^2 ~~~x\in [0,1]-P\end{cases}$$I want to calculate$$\int_0^1 f(x) dx$$What I try: I think the measure of Cantor set is zero,...
View ArticleIf $(a_n)\subset[0,\infty)$ is non-increasing and $\sum_{n=1}^\infty a_n
I'm studying for qualifying exams and ran into this problem.Show that if $\{a_n\}$ is a nonincreasing sequence of positive realnumbers such that $\sum_n a_n$ converges, then $\lim\limits_{n \rightarrow...
View ArticleComplete metric subspaces of $\mathbb{Q}$
Is there a nice characterization for the complete metric subspaces of $\mathbb{Q}$ (with the usual metric)? It seems like a such a subspace must have empty interior; if it contained an open interval...
View ArticleOpposite of a Dedekind cut
I'm reading the construction of $\mathbb{R}$ using Dedekind cuts on $\mathbb{Q}$. In my course note, a cut is a nonempty subset $A$ of $\mathbb{Q}$ satisfying:if $x \in A$ and $y \leq x$ then $y \in...
View ArticleThe constant in Gronwall's inequality
The classical Gronwall's inequality is as follows:Assume that$$f(t)\leq K+\int_0^tf(s)g(s)ds,$$where $f(t)$ and $g(t)$ are continuous functions in $[0,T],$$g(t)\geq 0$ for $t\in [0,T],$ and $K\geq 0.$...
View Articlesequence of integral of a function
Let $a_n=\frac{1}{n}\int_{0}^{n}\frac{log(2+x)}{\sqrt{1+x}}dx$ then i want to show $\lim_{n\to\infty}a_n\rightarrow 0$.i assume $b_n=\int_{n-1}^{n}\frac{log(2+x)}{\sqrt{1+x}}dx$ and...
View ArticleInfinite Integration Issues
Let me preface this by saying that I know I'm doing something wrong, I'm just here to find out what exactly.We know that integrating a function yields constants. Something like $e^x$ yields $+c$ when...
View ArticleHow many blocks of even and odd numbers will accrue in a cycle for the sum of...
Context: In this question I asked about "Finding a polynomial function for alternating $m$ numbers of odds and even numbers in sequences" I noticed that all Faulhaber polynomials are solutions for...
View ArticleWe know that $g \in L_1$ and that $g*g \leq g$ nearly everywhere. Show that:...
$g \in L_1$ such that $g*g \leq g$ nearly everywhere. Show that:$$\int_{\mathbb{R}} g(x) d\lambda_1(x) \leq 1$$From what is said, I know that:$$g*g \leq g \implies \int_{\mathbb{R}} (g*g) (x) \...
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Question on Theorem 1.11 in Baby Rudin
Definition 1.7 in Baby Rudin says: Suppose S is an ordered set, and E $\subset$ S. If there exists a $\beta \in$ S such that $x \leq \beta$ for every $x \in E$, we say E is bounded above and call...
View ArticleLet $BAC$ be a triangle. What can be said about the limiting slope of BC when...
QuestionLet $BAC$ be a triangle. Let a line $L$ of slope $m$ pass through the point $A (z,f(z))$ which is tangent of $f$ at the $x$ coordinate of $A$. Now assume $B$ and $C$ are moving such that $B$...
View Articlehow to understand the epsilon-nets are constructed in the followsing set...
I am reading an article about constructing nets on a set, but I do not fully understand how the epsilon-nets are constructed. The general idea is to partition the size of the coordinates of a vector...
View ArticleConvergence-determining class is a separating class
When I was reading Billingsley's book "Convergence of probability measures", it is claimed that "A convergence-determining class is obviously a separating class". But I don't understand why it's...
View ArticleJustification for inequality
Suppose $(M,g)$ is a Riemannian manifold with $\text{diam } M \leq D$. Let $u : M \rightarrow [0,\infty]$ be smooth. Define $R_i := 2^{-i}D$ so that $B_i := B(x,R_i)$ for a fixed $x \in M$. In...
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