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illysial
  • Member for 8 years, 8 months
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28 votes
4 answers
28k views

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

13 votes
1 answer
484 views

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

9 votes
3 answers
853 views

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

8 votes
1 answer
2k 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.

6 votes
1 answer
312 views

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

5 votes
1 answer
2k views

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

5 votes
3 answers
541 views

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

5 votes
1 answer
202 views

Question about the definition of the dual map of the differential

4 votes
0 answers
332 views

Understanding the construction of Exterior Algebra

4 votes
1 answer
972 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

4 votes
1 answer
317 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 $\bigcap \ker f_i \subset \ker f$

4 votes
1 answer
174 views

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

4 votes
2 answers
106 views

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

3 votes
1 answer
578 views

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

3 votes
4 answers
2k views

Fixed point for a continuous function on a compact set?

3 votes
3 answers
227 views

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

3 votes
1 answer
344 views

Counting Periodic Orbits on a regular Hexagon

3 votes
1 answer
3k views

L'hopital's Rule in higher dimensions.

3 votes
0 answers
74 views

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

3 votes
1 answer
434 views

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

3 votes
1 answer
1k views

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

3 votes
2 answers
493 views

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

3 votes
1 answer
64 views

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

2 votes
1 answer
51 views

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

2 votes
0 answers
197 views

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

2 votes
1 answer
49 views

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

2 votes
2 answers
75 views

Prove that if $f$ is linear, then for any $\textbf{a},\textbf{v} \in \mathbb{R}^2$, $f(\textbf{a}+\textbf{v})=f(\textbf{a})+[Df(\textbf{a})]\vec{v}$

2 votes
1 answer
114 views

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

2 votes
1 answer
135 views

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

2 votes
1 answer
389 views

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