Articles of real analysis

Maximal functions where weak type inequality fails

Let $\mathcal{R}$ denote the set of all open rectangles in $\mathbb{R}^2$ with sides parallel to the coordinate axis. Given a function on $\mathbb{R}^2$, consider the maximal function: $$f_{\mathcal{R}}^*(x)=\sup_{x\in R\in \mathcal{R}}\frac{1}{|R|}\int_R|f(y)|dy,\enspace x\in \mathbb{R}^2 ,$$ Problem the goal of this problem is to show that $f\mapsto f_{\mathcal{R}}^*$ does not satisfy the weak type inequality $$\left| \left\{x\in \mathbb{R}^2:\enspace […]

Does $G_{\delta}+q$ sets cover $\Bbb{R}$ a.e

Let $G_{\delta}$ be countable intersections of given open sets with positive Lebesgue measure on $[a,b]$. My question is that if $G_{\delta}+q$ covers $\Bbb{R}$ a.e, i.e. is $$ \bigcup_{q \in \mathbb{Q}}(q+G_{\delta})=\Bbb{R}-N $$ true? ($N$ is of Lebesgue measure zero). $G_{\delta}$ must be uncountable for it has positive Lebesgue measure. But it may has empty interior. I […]

Prove the following theorem involving homeomorphic metric spaces.

I want to prove this theorem: If $(X,d)$ and $(X,d’)$ be homeomorphic metric spaces, then they have the same convergence sequences. However, there exists homeomorphic metric spaces $(X,d), (X,d’)$ such that only one of them is complete. So what we want to prove is that if $d(x_n,x)< \epsilon$ then $d'(x_n,x)<\epsilon$ but the thing is that […]

Computing the trigonometric sum $ \sum_{j=1}^{n} \cos(j) $

I have a task to compute such a sum: $$ \sum_{j=1}^{n} \cos(j) $$ Of course I know that the answer is $$ \frac{1}{2} (\cos(n)+\cot(\frac{1}{2}) \sin(n)-1) = \frac{\cos(n)}{2}+\frac{1}{2} \cot(\frac{1}{2}) \sin(n)-\frac{1}{2} $$ but I don’t have any idea how to start proving it.

Prove that Open Sets in $\mathbb{R}$ are The Disjoint Union of Open Intervals Without the Axioms of Choice

There are several proofs I have seen of this, but they all seem to use choice subtely at some point. Is there any way to prove this without choice, or is it possibly unproveable?

The importance of continuity of functions in proving some properties of convolution.

What is the importance of continuity of functions in proving the commutative and associative properties of convolution if the functions are assumed to be Riemann integrable $2\pi$-periodic functions ? Could anyone clarify this for me? As the book “Fourier Analysis” by Stein and Shakarchi assumes that the functions are continuous at the beginning of the […]

Properties of a convex set

Let $C \subset \mathbb{R}^n$ be convex and closed, $y \in \partial C$. Show that there is an affine $f(x)=\langle x-y,e \rangle$ where $|e|=1$, such that $f(y)=0$ and $\forall x \in C: f(x) \leq 0$ Notation: Let me explain my notation, in case there is some confusion: $\langle -,- \rangle$ is the euclidean scalar product and […]

Is $\sup_{E\ \text{of finite measure}}\mu(E)<+\infty$ equivalent to $\mu(X)<+\infty$ in $(X,\Sigma,\mu)$?

This relates to a previous question I posted. Let $(X,\Sigma,\mu)$ be a measure space with $\mu(X)<+\infty$. Then $\sup_{E\in{\mathcal A}_{\infty}}\mu(E)<+\infty$, where $ {\mathcal A}_{\infty}=\{A\in\Sigma:\mu(A)<+\infty\}. $ My question is: Is there a measure space such that $\sup_{E\in{\mathcal A}_{\infty}}\mu(E)<+\infty$ but $\mu(X)=+\infty$?

Vector space or vector field?

I seem to be having a problem distinguishing between a vector space (which I know to be a set of vectors over some scalar set) and a vector field. I know that in Multivariable Calculus a vector field is a vector-valued function (being a derivative for conserved fields.) I feel they are different but how […]

Characterizing the continuum using only the notion of midpoint

Is it possible to categorically characterize the continuum using only the notion of midpoint? I assume some kind of Dedekind-cut-axiom will be necessary, but I don’t know how to define order (see my related question).