Here is a statement in Zygmund’s Measure and Integral that confuses me on Page 8: Any closed set in $\mathbb{R}^1$ can be constructed by deleting a countable number of open disjoint intervals from $\mathbb{R}^1$. A perfect set results by removing the intervals in such a way as to create no isolated points; thus, we would […]

Let $f, g$ be measurable real-valued functions on $[0, \infty)$, with $$\int_0^{x} g \leq \int_0^x f$$ for each $x$. Show that if $\phi: [0, \infty) \rightarrow [0, \infty)$ is nonincreasing, then also $$\int_0^{\infty} \phi g \leq \int_0^{\infty} \phi f$$ I just got no idea where to start. I know that $\phi$ must decrease to a […]

This question already has an answer here: Can there be two distinct, continuous functions that are equal at all rationals? 4 answers

Assume $f(x)$ is continuous on $[a, b]$. (a) Prove that if $f(x) > 0$ for all $x \in [a, b]$, then the integral of $f(x)$ from $a$ to $b > 0$. I tried to apply the Extreme Value Theorem and into it, but I can not deal with it successfully. (b) Prove that if $f(x) […]

Let $f:\mathbb{R}^+\to\mathbb{R}^+$ and $g:\mathbb{R}^+\to\mathbb{R}^+$ be two strictly concave, strictly increasing, twice differentiable functions, such that $f(x)=O(g(x))$ as $x\to\infty$, i.e. there exists $M>0$ and $x_0$ such that $$f(x)\leq Mg(x)\qquad \forall x\geq x_0.$$ Is it true that $f'(x)=O(g'(x))$ as $x\to\infty$? (this is an extensions of this question)

Let $\{f_k(x)\}_{k=0}^\infty$ be a Littlewood-Paley decompositon, that is, $$ f_k \in C_c^\infty $$ $$ \sum_{k=0}^\infty f_k (x) = 1,$$ $$ \text{supp} f_0 \subset \{ |x| \leq 2 \},$$ $$ \exists f \in C_c^\infty \; \text{such that}\; \text{supp} f \subset \{ 2^{-1} \leq |x| \leq 2 \} \; \text{satisfying}\; f_k (x) = f(x/2^k).$$ Then I hope […]

For $0 \le x \le 1 , p\gt 1$, prove $1 \over2^{p-1}$ $\le x^p +(1-x)^p \le 1$.

I am familiar with two different sets of field axioms. The first one is from “Mathematical Analysis” by Apostol. It has the first $3$ usual axioms, but the $4^{th}$ one is different: Axiom 1: Commutative Laws $x+y=y+x$, $xy=yx$ Axiom 2: Associative Laws $x+(y+z)=(x+y)+z$, $x(yz)=(xy)z$ Axiom 3: Distributive Law $x(y+z)=xy+xz$ Axiom 4: Given any two real […]

We say that $f$ is of bounded variation over $[a,b]$ or $f\in BV[a,b]$ if $V_f[a,b] = \sup \sum_{k=1}^n |f(x_k) – f(x_{k-1})| < \infty, $ where the supremum is taken over all possible partitions of $[a,b]$ My question is: If $f \in BV[a,b],$ show that $|f(x)|\leq |f(a)| + V_f[a,b] \ \ $ for all $x\in [a,b],$ […]

I was curious about this exercise, because I thought it could be a valuable tool to use the theorem dell’asintoto … I do not think that is the way, does anyone have any idea? ! Let $ h $ is a function defined on $(a, + \infty)$ and limited all intervals $(a, b)$ $a <$ […]

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