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6 votes
2 answers
325 views

Is the hypothesis $\mu \ge 0$ redundant in Brezis's Ex 3.15?

5 votes
0 answers
181 views

Is this a typo in Brezis's Ex 3.24?

5 votes
0 answers
111 views

If $f_n \overset{\star}{\rightharpoonup} f$ in $\sigma(E^\star, E)$, then $\|f\| \le \liminf \|f_n\|$

5 votes
0 answers
118 views

If $f$ is real-valued bounded measurable and $\mu$ a complex measure, then $\left | \int_X f \mathrm d \mu \right | \le \int_X |f| \mathrm d |\mu|$

5 votes
2 answers
182 views

Is it necessary to consider the case $p=1$ separately?

5 votes
0 answers
88 views

Determine the orthogonal projection of $f$ onto $K := \{u \in H : |u| \le h \text{ a.e.}\}$

5 votes
1 answer
86 views

If $(t_n u_n-t_m u_m)(u_n-u_m) \le 0$ for all $m, n \in \mathbb N$, then $(u_n)$ is convergent

4 votes
2 answers
81 views

Prove that if $t:=\lim_n t_n >0$, then $(u_n)$ converges

4 votes
0 answers
50 views

If $X$ is infinite-dimensional normed vector space, then $\mathcal C_c(X) = \{0\}$

4 votes
1 answer
92 views

Counter-example: approximate an integrable function from below by a continuous function with compact support

4 votes
2 answers
104 views

A good upper bound of $S_n := \sum_{k=1}^n e^{-\lambda} \frac{\lambda^k}{k!} \frac{1}{(k \varepsilon)^k}$ in terms of $\varepsilon,\lambda,n$

4 votes
0 answers
73 views

A proof that $\sum_{i=0}^{p_n-1} f(Y_{t_i^n})(X_{t_{i+1}^n}-X_{t_i^n})^2$ converges a.s. to $\int_0^t f(Y_s)\mathrm{d}\langle X, X\rangle_s$

4 votes
2 answers
498 views

The source of this excellent lecture note on Brownian motion

4 votes
2 answers
361 views

Let $(M_t)$ be a continuous square-integrable martingale with independent increments. Is $t \mapsto \mathbb E[M_t^2]$ continuous?

4 votes
1 answer
180 views

Let $E$ be a Banach space, $p \in (1, \infty)$, and $L_p := L_{p}(X, \mu, E)$. Is $(L_p)^* \cong L_{p'}$ where $\frac{1}{p} + \frac{1}{p'} = 1$?

4 votes
1 answer
101 views

Hausdorff topological spaces: Is the pointwise limit of a sequence of Borel measurable functions again Borel measurable?

4 votes
2 answers
343 views

How to prove that $\mu_n \rightharpoonup \mu$ IFF $\mu_n \overset{*}{\rightharpoonup} \mu$ and $\mu_n (X) \to \mu (X)$?

4 votes
1 answer
217 views

Where does my proof of Milman-Pettis theorem break down?

4 votes
0 answers
135 views

A proof without using net in Brezis's Ex 3.14

4 votes
0 answers
41 views

Let $\mu, \nu$ both have finite second moments and $\nu \in \Pi(\mu, \nu)$. Then the inner product $\langle \cdot, \cdot\rangle$ is $\pi$-integrable

3 votes
1 answer
84 views

The map $\varphi : E \to \mathbb R \cup \{+\infty\}$ is proper convex l.s.c. if and only if $\varphi = \varphi^{**}$

3 votes
1 answer
125 views

It is still true that $\dim X \le \dim X^\star$ for infinite-dimensional Banach spaces if $\dim X < 2^{\aleph_0}$?

3 votes
1 answer
76 views

Finite signed measures: reconcile different types of convergence

3 votes
1 answer
252 views

An exposition of Tao's proof of disintegration theorem

3 votes
0 answers
75 views

How to prove these definitions of an analytic set are equivalent?

3 votes
1 answer
247 views

The Borel $\sigma$-algebra generated by the product topology coincides with the product of Borel $\sigma$-algebras: where did I get wrong?

3 votes
1 answer
287 views

What does it mean for a signed measure to be "regular" in Riesz-Markov-Kakutani theorem?

3 votes
0 answers
371 views

$3$ versions of Riesz–Markov–Kakutani theorem

3 votes
0 answers
221 views

If $\mu$ is atomless then its c.d.f. is continuous

3 votes
0 answers
80 views

If $\mu(B_n) \to 1$ then $\int_{B_n} f \mathrm d \mu \to \int_{X} f \mathrm d \mu$

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