User illysial

15

18

votes

3

answers

12k

views

The closure of a connected set in a topological space is connected

general-topology connectedness

May 30 at 17:44 Stensrud 55

2

11

votes

1

answer

277

views

Will the Lebesgue integral of a real valued function always be a Riemann sum?

real-analysis measure-theory

Sep 4 '15 at 14:06 PhoemueX 27.9k

7

votes

1

answer

769

views

If $\mu$ is finite, then $\{f_n\}$ is uniformly integrable iff $\sup_n \int|f_n| d\mu<\infty$ and $\{f_n\}$ is uniformly absolutely continuous.

measure-theory proof-verification lebesgue-integral

Oct 8 at 23:35 Robson 773

2

6

votes

1

answer

102

views

A collection $\{f_\alpha\}_{\alpha \in A}$ so that $\sup_{\alpha \in A} f_{\alpha}(x)$ is finite and non-measurable

measure-theory lebesgue-measure

Jul 16 '15 at 5:08 Math Wizard 13.5k

2

5

votes

2

answers

390

views

Prove that $\lim_{t\to \infty} t\mu(\{x:f(x)\geq t\})=0$

real-analysis integration measure-theory

Jun 24 '15 at 8:54 Adam Hughes 32.5k

4

votes

0

answers

175

views

Understanding the construction of Exterior Algebra

linear-algebra abstract-algebra differential-geometry tensor-products multilinear-algebra

Apr 3 '16 at 2:39 darij grinberg 12k

1

4

votes

1

answer

101

views

Question about the definition of the dual map of the differential

differential-geometry

Mar 19 '16 at 14:18 Fallen Apart 2,161

2

4

votes

1

answer

131

views

Let $f_1,f_2,\ldots, f_n$ be linear functionals on $X$. Show $f=\sum_{i=1}^n\lambda_i f_i$ iff $\bigcap \ker f_i \subset \ker f$

functional-analysis

Jul 19 '15 at 0:20 Theo Bendit 24.1k

4

votes

1

answer

403

views

Suppose $\mu$ is a finite measure and $\sup_n \int |f_n|^{1+\epsilon} \ d\mu<\infty$ for some $\epsilon$. Prove that $\{f_n\}$ is uniformly integrable

measure-theory proof-verification lebesgue-integral

Jun 21 '15 at 5:26 d.k.o. 10.6k

4

votes

1

answer

122

views

Find the exact value of $t$ that maximizes $\int_{t}^{t+1}\sin e^x \ dx$

calculus

May 6 '14 at 16:49 Eugene Bulkin 1,027

3

votes

1

answer

679

views

Show that $\Gamma(z)\Gamma(1-z) \sin \pi z$ is bounded in the complex plane

complex-analysis gamma-function beta-function

Mar 1 '16 at 6:05 Mhenni Benghorbal 43.6k

3

votes

1

answer

53

views

Prove $\int_0^{t_1}\int_0^{\tau}f(s)\ ds\ d\tau=\int_0^{t_1}(t_1-s)f(s)ds$

calculus integration

Sep 9 '15 at 2:14 illysial 1,678

2

3

votes

3

answers

294

views

What is wrong with my application of Lebesgue Dominated Convergence Theorem in these two examples?

measure-theory lebesgue-integral

Jun 18 '15 at 12:59 Alex M. 30.1k

3

votes

0

answers

60

views

Given a point $A$, describe those points to which a catenary cannot be drawn from $A$.

calculus calculus-of-variations

Sep 17 '14 at 0:01 illysial 1,678

3

votes

4

answers

1k

views

Fixed point for a continuous function on a compact set?

real-analysis compactness fixed-point-theorems

Aug 9 '14 at 10:00 Martin Sleziak 46.1k

3

votes

1

answer

324

views

Solve $3^x+3^{(3x+1)}=108$

algebra-precalculus

Aug 8 '14 at 1:33 k170 7,519

3

votes

2

answers

73

views

Sufficient Condition for $\lim_{n\rightarrow \infty} \frac{a_n}{b_n}=1$

real-analysis

Aug 5 '14 at 1:22 RRL 56.6k

2

3

votes

1

answer

2k

views

L'hopital's Rule in higher dimensions.

multivariable-calculus

Jul 13 '14 at 2:58 Peter Franek 7,391

3

votes

1

answer

203

views

Counting Periodic Orbits on a regular Hexagon

combinatorics geometry billiards

Apr 5 '14 at 20:11 Alex Becker 49.7k

3

votes

3

answers

189

views

A function $F:\mathbb{Q} \rightarrow \rm numerators$

number-theory

Apr 5 '14 at 0:08 Hurkyl 113k

1

2

votes

2

answers

203

views

Using Universal Mapping Property of Tensors to show that $\dim V \otimes W=(\dim V)(\dim W)$

linear-algebra abstract-algebra tensor-products

Apr 2 '16 at 17:33 egreg 189k

2

votes

1

answer

43

views

If $\mathrm{min}_{u\in U}||v-u||=||v-u_0||$, then $v-u_0\in U^{\perp}$

linear-algebra inner-product-space

Sep 6 '15 at 0:52 illysial 1,678

2

votes

0

answers

138

views

$f_n \rightarrow 0$ weakly if and only if $(\|f_n\|)_{n=1}^{\infty}$ is bounded and $f_n$ converges pointwise to $0$.

functional-analysis measure-theory lebesgue-integral

Jul 10 '15 at 1:12 Math Wizard 13.5k

2

votes

1

answer

181

views

Concerning existence of subsequence of converging integrals on subsets of $[0,1]$ of a sequence $(f_n)\in[0,1]$

measure-theory lebesgue-integral

Jun 19 '15 at 4:56 illysial 1,678

2

votes

1

answer

273

views

Find the limit $\lim_{n\to\infty} \int_0^n\left(1-\frac{x}{n}\right)^n\log(2+\cos(x/n)) \ dx$

measure-theory lebesgue-measure

Jun 17 '15 at 21:41 Community♦ 1

1

2

votes

1

answer

620

views

Composition of 2 Lebesgue measurable functions is not lebesgue measurable: Are these two functions Borel Measurable?

measure-theory

Jun 14 '15 at 3:10 PhoemueX 27.9k

1

2

votes

1

answer

210

views

If $f=g$ on $(x-r,x+r)\cap (0,1)$ and $g$ is Borel measurable, then $f$ is Borel measurable.

measure-theory

Jun 11 '15 at 18:16 copper.hat 130k

1

2

votes

1

answer

72

views

If $X=\{0,1\}$, there exists an outer measure $\mu^$ on $X$ such that $\mu^ \neq \mu^+$

real-analysis measure-theory

May 24 '15 at 22:24 Lost in a Maze 1,049

2

votes

1

answer

80

views

Show that $g(x)=\text{mid}\{f_1,f_1,f_3\}$ is measurable.

real-analysis

Apr 14 '15 at 16:44 Cameron Buie 89.4k

2

votes

1

answer

40

views

How is the Chain rule used to show that $\int_t^{t+\tau} f(x(t))\frac{dx(t)}{dt} \ dt=\int_{x(t)}^{x(t+\tau)} f(x) \ dx$?

calculus

Nov 19 '14 at 23:53 user795305 332

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52 Questions

The closure of a connected set in a topological space is connected

Will the Lebesgue integral of a real valued function always be a Riemann sum?

If $\mu$ is finite, then $\{f_n\}$ is uniformly integrable iff $\sup_n \int|f_n| d\mu<\infty$ and $\{f_n\}$ is uniformly absolutely continuous.

A collection $\{f_\alpha\}_{\alpha \in A}$ so that $\sup_{\alpha \in A} f_{\alpha}(x)$ is finite and non-measurable

Prove that $\lim_{t\to \infty} t\mu(\{x:f(x)\geq t\})=0$

Understanding the construction of Exterior Algebra

Question about the definition of the dual map of the differential

Let $f_1,f_2,\ldots, f_n$ be linear functionals on $X$. Show $f=\sum_{i=1}^n\lambda_i f_i$ iff $\bigcap \ker f_i \subset \ker f$

Suppose $\mu$ is a finite measure and $\sup_n \int |f_n|^{1+\epsilon} \ d\mu<\infty$ for some $\epsilon$. Prove that $\{f_n\}$ is uniformly integrable

Find the exact value of $t$ that maximizes $\int_{t}^{t+1}\sin e^x \ dx$

Show that $\Gamma(z)\Gamma(1-z) \sin \pi z$ is bounded in the complex plane

Prove $\int_0^{t_1}\int_0^{\tau}f(s)\ ds\ d\tau=\int_0^{t_1}(t_1-s)f(s)ds$

What is wrong with my application of Lebesgue Dominated Convergence Theorem in these two examples?

Given a point $A$, describe those points to which a catenary cannot be drawn from $A$.

Fixed point for a continuous function on a compact set?

Solve $3^x+3^{(3x+1)}=108$

Sufficient Condition for $\lim_{n\rightarrow \infty} \frac{a_n}{b_n}=1$

L'hopital's Rule in higher dimensions.

Counting Periodic Orbits on a regular Hexagon

A function $F:\mathbb{Q} \rightarrow \rm numerators$

Using Universal Mapping Property of Tensors to show that $\dim V \otimes W=(\dim V)(\dim W)$

If $\mathrm{min}_{u\in U}||v-u||=||v-u_0||$, then $v-u_0\in U^{\perp}$

$f_n \rightarrow 0$ weakly if and only if $(\|f_n\|)_{n=1}^{\infty}$ is bounded and $f_n$ converges pointwise to $0$.

Concerning existence of subsequence of converging integrals on subsets of $[0,1]$ of a sequence $(f_n)\in[0,1]$

Find the limit $\lim_{n\to\infty} \int_0^n\left(1-\frac{x}{n}\right)^n\log(2+\cos(x/n)) \ dx$

Composition of 2 Lebesgue measurable functions is not lebesgue measurable: Are these two functions Borel Measurable?

If $f=g$ on $(x-r,x+r)\cap (0,1)$ and $g$ is Borel measurable, then $f$ is Borel measurable.

If $X=\{0,1\}$, there exists an outer measure $\mu^$ on $X$ such that $\mu^ \neq \mu^+$

Show that $g(x)=\text{mid}\{f_1,f_1,f_3\}$ is measurable.

How is the Chain rule used to show that $\int_t^{t+\tau} f(x(t))\frac{dx(t)}{dt} \ dt=\int_{x(t)}^{x(t+\tau)} f(x) \ dx$?