Theoretical foundations of calculus: limits, convergence of sequences, construction of the real numbers, least upper bound property, and related analysis topics such as continuity, differentiation, and integration through the fundamental theorem of calculus.

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How much larger is the $\sigma$-algebra than the algebra in Caratheodory extension?

Given a 'measure' $\lambda$ on an algebra $\mathcal{A}$ of sets, Caratheodory gives a way to extend this $\lambda$ to a $\sigma$-algebra. The idea is we define an outer measure (on all subsets) ...
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90 views

Is the range of the Fourier transform $L^1(\mathbb{R}) \to C_0(\mathbb{R})$ closed under “quasi-inversion”?

Definitions: A function $f \in C_0(\mathbb{R})$ is quasi-invertible if $1 \notin \operatorname{ran}(f)$. The quasi-inverse of such an $f$ is $\frac{f}{f-1}$. Some discussion: Let $C_1(\mathbb{R})$ ...
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56 views

representation of points of continuity of a function $f :\mathbb{R}\rightarrow \mathbb{R}$

Question is : Suppose $f$ is continuous at $x\in \mathbb{R}$ we need : for given $\epsilon >0 $ existence of $\delta > 0$ such that $|x-y|< \delta$ implies $|f(x)-f(y)|< \epsilon$ ...
4
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53 views

Zero arithmetic mean: bound on Abel sum?

Let $(a_0, a_1, a_2, \ldots)$ be a bounded sequence in $\mathbb{R}$ with arithmetic means \begin{equation} \frac{a_0 + a_1 + \cdots + a_{n-1}}{n} \end{equation} converging to zero as $n \to \infty$. ...
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60 views

Doubt about computability of integrals over open sets.

In Spivak's Calculus on Manifolds after showing that partitions of unity exists, Spivak defines integrals of functions over open sets as follows. He says: "An open cover $\mathcal{O}$ of an open ...
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66 views

Computing volume of ball in $n$ dimensions

Let $B^n(a)$ denote the closed ball of radius $a$ in $\mathbb{R}^n$, centered at $0$. Show that $v(B^n(a))=\lambda_n a^n$ for some constant $\lambda_n$, where $v(X)$ denotes the volume of $X$. By ...
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133 views

Lower semicontinuous and discontinuous everywhere real bounded function?

Does there exist an $f:\mathbb{R}\rightarrow\mathbb{R}$ that is bounded such that for any $a$ then $f^{-1}(a,+\infty)$ is open but $f$ is discontinuous everywhere? Such a function seems too likely to ...
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84 views

Does Weak $L^{2}$ Convergence on Finite Measure Spaces Imply Strong $L^{1}$ Convergence?

When $\mu(X)<\infty$, $L^{1}(\mu)\supset L^{2}(\mu)$ and by the Riesz-Fischer theorem, weak convergence of $f_{n}\to f$ in $L^{2}$ is equivalent to ...
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31 views

Fubini lookalike for arbitrary set

Let $A$ and $B$ be rectangles in $\mathbb{R}^k$ and $\mathbb{R}^n$ respectively. Let $S$ be a set contained in $A\times B$. For each $y\in B$, let $$S_y=\{x\mid x\in A\text{ and }(x,y)\in S\}.$$ We ...
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139 views

Sequence of convex functions converges uniformly

I am working on the following problem. Let $f_{n}: [a, b] \rightarrow \mathbb{R}$ be a sequence of convex functions. Furthermore, for each fixed $x \in [a, b]$, suppose $f(x) = \lim_{n ...
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38 views

On the existence of a weight function making sequence of integration preserving limit

The problem goes as follows: Let $f_n$ be strictly positive Lebesgue measurable function defined on $[0,\infty)$ satisfying $$\lim_{n \to \infty} \int_0^\infty f_n(x)\ dx=0$$ then show that there ...
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176 views

Simple problem of a differentiable function

Please, can somebody help me with this problem? I tried to use the Mean Value Theorem, but couldn't solve it. Let $g: [a,b]\rightarrow\mathbb{R}$ a differentiable function on $[a,b]$. If ...
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338 views

Riemann-Stieltjes Integral computation (step function?)

Im trying to integrate this, using theorem 7.9 of apostol's book: $$\int^{10}_0 f(x)d\alpha(x) $$ $f(x) = x^2$ and $\alpha(x)= 3\chi(7,9](x)$ Where $\chi(x)$ is $0$ everywhere except $1$ in the ...
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309 views

Bartle or Cohn as a first text in Measure Theory.

Consider the following texts: (a) Robert Bartle - Elements of Integration and Lebesgue Measure (b) Donald Cohn - Measure Theory. Which text would you recommend for a student with only a modest ...
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617 views

Strict Inequality in Rudin's Proof of the Riesz Representation Theorem

In Rudin's proof of the Riesz Representation Theorem (step ten), he proves that $$\Lambda h_i \leq \mu(V_i) < \mu(E_i) + \epsilon/n , \quad \mu(K) \leq \sum\limits_{1 \leq i \leq n} \Lambda h_i.$$ ...
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146 views

Properties of the projection onto a nonconvex set

Consider a set $\Omega\subseteq\mathbb{R}^n$ being "sufficiently regular", for example being the image of a $C^1$ mapping from $\mathbb{R}^p$ for some $p\ge1$. We may then consider the mapping $$ ...
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75 views

Constructivism implied or not

Let me take up some details in the answer of another question. Submitted by user hyg17: Heading: All real numbers can be expressed as a limit of rational numbers? The question was: Let $C$ be a set ...
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204 views

Lebesgue measure as a fixpoint: change of variables formulas

This question is inspired by several others on a similar topic: see e.g. this one and a sequence of linked questions. Let us so far focus on $\Bbb R^n$ endowed with standard Borel structure. For any ...
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178 views

Stokes' Theorem problem

Let $M \subset \mathbf{R}^n$ be oriented compact smooth $k$-manifold and $\alpha$ be a $C^1$ diferential $(k-1)$-form defined in a neighborhood of M. Use Stokes' theorem to prove that \begin{align*} ...
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88 views

Possible error in Wood's report on polylogarithms

I'm studying the article by Wood, D.C. "The Computation of Polylogarithms. Technical Report 15-92*" PS (it is remarkably poorly translated from latex to ps). It is listed in the literature section on ...
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84 views

$f(a+b,\lambda) = f (a,\lambda) \cdot f(b,\lambda)$

Consider the equation: $f(a+b,\lambda) = f (a,\lambda) \cdot f(b,\lambda)$, for $a \geq 0$ and $b \geq 0$. Is my understanding that this simple functional equation is important in analysis. Can ...
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2k views

Proving rigorously the supremum of a set.

Suppose $A \subset \mathbb{R} \neq \emptyset$. Let $A = [\,0,2).\,\,$ Prove that $\sup A = 2$ Attempt: $A$ is the half open interval $[\,0,2)$ and so all the $x_i \in A$ look like $0 \leq x_i < ...
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85 views

Partition of $\mathbb{R}$ into nullset and 1st category set

Let $\{a_i\}_{i=1}^{\infty}$ be an enumeration of the rationals, $\mathbb{Q}$. Let $I_{ij}$ be the open interval centered at $a_i$ and having length $1/2^{i+j}$, and define $G_j = \cup_{i=1}^\infty ...
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587 views

Infinite self-convolution for a function

I have a mathematical problem that leads me to a particular necessity. I need to calculate the convolution of a function for itself for a certain amount of times. So consider a generic function $f : ...
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73 views

$x\mapsto f(x^{1/p})$ is smooth if and only if $f^{(n)}(0)=0$ whenever $p\nmid n$

This is a slight generalization of something I got stuck on when trying to do Problem 2-5 from Introduction to Smooth Manifolds by John M. Lee (which uses the case $p=3$). Let $p\ge 1$ be an ...
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215 views

interchange limits (not a sequence of function)

I am aware of a theorem in analysis regarding the result that $$\lim_{t\rightarrow x} \lim_{n\rightarrow \infty} f_n(t) = \lim_{n\rightarrow \infty} \lim_{x\rightarrow t} f_n(t).$$ My question is ...
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304 views

An absolutely continuous cumulative distribution function that fails to have a Riemann-integrable pdf.

We know that if a real-valued random variable $ X $ on a probability space has an absolutely continuous cumulative distribution function (cdf) $ F $, then $ X $ possesses a probability density ...
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244 views

Second derivative of a vector field

I wonder how to treat the "second derivative" of a vector field. For example, imagine we have a vector field $f:\mathbb{R}^n \rightarrow \mathbb{R}^n$. Then we evaluate the derivative at two points ...
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438 views

Help me correct these properties of : $f_{n}(x)= nx(1-x)^{n}$? Is there maybe a typo in the sequence?

Examine the sequence of functions $(f_n)_{n\in \mathbb{N}}$ on $x\in[0,1]$: $$f_{n}(x)= nx(1-x)^{n}$$ Does $(f_n)_{n\in \mathbb{N}}$ converge pointwise or uniform? I will show that it does ...
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90 views

The Limit of the Following Derivative

Suppose you have two functions $F$ and $G$ with the following properties. $G(0)=F(0)=0, G'>0, G''<0, F'>0, F''<0 $ and also $\lim_{x\to0} F'(x)=\infty, \lim_{x\to\infty} F'(x)=0, ...
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171 views

Constructing the support of a Borel measure

From Rudin, Real and Complex Analysis, Chapter 8, Problem 7, 1st Edition. Suppose $E$ is a compact set in $\mathbb{R}^{k}$ without isolated points. Show that $E$ is the support of a continuous ...
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132 views

When does $|f*g|_{p}=|f|_{1}|g|_{p}$?

From Rudin, Real and Complex Analysis, 1st edition, Chapter 7, Problem 4 Suppose $1\le p\le \infty$, $f\in L^{1}(\mathbb{R}^{1})$, $g\in L^{p}(\mathbb{R}^{1})$. Show that the the integral defining ...
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163 views

Inexact Newton method.

Let's a nonlinear function $ f:[-1,+1]^N\subset\mathbb{R}^N\to\mathbb{R}^N,\; N\in\mathbb{N}, $ such that the the sequence generated by the method of Newton-Raphson $$ x_{n+1}=x_n-[Df(x_n)]^{-1}\cdot ...
4
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160 views

Riemann Integrable $f$ and Real Analysis Proofs

I am solving old comprehensive real analysis exams and there are two questions that I can not be sure, If $f$ is Riemann integrable then $|f|^r$ is Riemann integrable for any $r>0$.( True or ...
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159 views

Functions from the Cantor set

Consider the Cantor set $\Delta\subset [0,1]$. Let $f\colon \Delta\to [0,\infty)$ be a continuous injection. Must $f$ be monotone on some uncountable closed subset of $\Delta$? Note that that van der ...
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2k views

prove that every continuous function is integrable

Can someone tell me whether this is correct thank you! We know that if a function f is continuous on $[a,b]$, a closed finite interval, then f is uniformly continuous on that interval. This means ...
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119 views

A question about functions in $L^p(E)$

I've been working on a problem. I found a couple of related questions on the site, but I was having a little bit of trouble clarifying everything. The goal is to show that given $f\notin L^p(E)$, ...
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185 views

Equivalent definition of lower semi-continuity

I have some problems about solving an exercise: Prove that one function $f \colon [a,b] \to \mathbb{R}$ is lower semi-continuous if and only if, for all $x \in [a,b]$, we have $$f(x)=\sup \{g(x) ...
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239 views

Prove that liminf of functions is semicontinuous

I recall that, if $\psi:\Bbb{R}\longrightarrow \Bbb{R}$ is a function defined over $\Bbb R$ with euclidean topology, we have $\liminf\limits_{y\to x} \psi(y) = \sup\limits_{U\in \mathscr{U}_x} ...
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144 views

How many points does one need for an epsilon-net

Does anyone know, how many points does one need to have an $\varepsilon$-net on a unit sphere sitting in the three-dimensional Euclidean space? Thanks!
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132 views

$C(K,R)$ continuous functions on compact metric spaces

How to do this problem? Question: let $(K,d)$ be a compact metric space and $A$ is contained in $C(K,R)$, the set of all continuous functions from $K$ to $\mathbb R$, an algebra that separates the ...
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166 views

Help with removing singularities involving $ \int_{1}^{\infty} \exp\left( - \frac{x^2}{2y^{2r}} - \frac{y^2}{2}\right) \frac{dy}{y^r}$

This post can be thought of as the prototype proof and the motivation for the question posted here Using a laplace type expansion to get bounds on an integral arising in the study of Brownian motion ...
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160 views

Using a laplace type expansion to get bounds on an integral arising in the study of Brownian motion

Let $ 0 < r < 1$, fix $x > 1$ and consider the integral $$ I_{r}(x) = \int_{1}^{\infty} \exp\left( - \frac{x^2}{2y^{2r}} - \frac{y^2}{2}\right) \frac{dy}{y^r}.$$ In the investigation of ...
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379 views

Interchanging the order of limits

Would you advise me on the references of Pringsheim Convergence about interchanging the order of limits? Where can I find the most general statement?
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786 views

First order variation and total variation of a function/stochastic process

The notions of first-order variation and total variation of a function or a stochastic process are equated in this book. However, I found their definitions different from two other sources: In ...
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136 views

Arclength integral

Suppose that $f: [a,b] \rightarrow \mathrm{R}^n$ is continuous with a derivative $f'$ whose norm is Riemann-integrable. To demonstrate the arclength integral formula, I'm trying to prove that, for ...
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440 views

Convergent sum with primes

If $f(n)$ is a strictly increasing elementary function from the reals to the reals, and $p(n)$ is the $n$'th prime number. Is there any $f(n)$ such that $\sum_{n=1}^\infty\frac{1}{f(p(n))}$ is ...
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2k views

Derivative of a differentiable everywhere function $f\colon \mathbb{R}\to\mathbb{R}$ is Borel measurable?

So my homework problem is to show that if a function $f: \mathbb{R} \to \mathbb{R}$ is differentiable everywhere, then its derivative $f'$ is Borel measurable. What I have for something to be ...
4
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172 views

Uniform convergence of piecewise linear interpolations

Let $$X^k (t) := X^0 (t+t_k) - X^0 (t_k)$$ where $X^0(t)$ is the piecewise linear interpolation of $X^0(t_k) \equiv X_k=\sum_{i=0}^{k-1} a_i b_i$ with interpolation intervals $a_k$. ...
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213 views

Equivalence of two sequences

I'm having some trouble showing that two things I really want to be the same are in fact the same. I want to show that these two sequences are, in fact, the same thing: $$a_0=1,a_1=-1, ...