Question

If $f_n \rightarrow f$ is a sequence of $L^p$ and $g_n \rightarrow g$ a bounded sequence of $L^{\infty}$ then $f_ng_n \rightarrow fg$ in $L^p$

score 392 · Answer 1 · 2026-04-15 12:07:05

392

Views

If $f_n \rightarrow f$ is a sequence of $L^p$ and $g_n \rightarrow g$ a bounded sequence of $L^{\infty}$ then $f_ng_n \rightarrow fg$ in $L^p$

Published on 15 Apr 2026 - 12:07

score 323 · Answer 2 · 2026-04-16 04:12:41

323

Views

Show that $\lim_{n\to \infty}\int_{0}^{1}g(x)f(nx)dx = \Big(\int_{0}^{1}g(x)dx\Big)\Big(\int_{0}^{1}f(x)dx\Big)$

Published on 16 Apr 2026 - 4:12

score 25 · Answer 3 · 2026-04-17 04:44:15

25

Views

Showing that if $K \in L_{loc}^{\infty}(\mathbb{R}^+)$ then $|K^{\star n}| \leq M(a)^n \frac{x^{n-1}}{(n-1)!}$ where $M(a)$ is the supremum on $[0,a]$

Published on 17 Apr 2026 - 4:44

score 114 · Answer 4 · 2026-04-11 17:52:43

114

Views

Can $\limsup = \liminf$ only almost surely?

Published on 11 Apr 2026 - 17:52

score 282 · Answer 5 · 2026-04-10 05:17:39

282

Views

Fubini's Theorem for Radon complex measures

Published on 10 Apr 2026 - 5:17

score 234 · Answer 6 · 2026-04-14 02:44:47

234

Views

Confusing of the last line of proof for Fatou's lemma

Published on 14 Apr 2026 - 2:44

score 94 · Answer 7 · 2026-04-17 10:24:53

94

Views

Existence of random Variable $X$ such that $\lim_{n\to\infty} \mathbb{E}[(X_n - X)^2] = 0$

Published on 17 Apr 2026 - 10:24

score 272 · Answer 8 · 2026-04-12 04:09:01

272

Views

Measure defined by integral is non-atomic

Published on 12 Apr 2026 - 4:09

score 69 · Answer 9 · 2026-04-13 13:29:52

69

Views

Is there a name for a bijection that is bi-measurable?

Published on 13 Apr 2026 - 13:29

score 127 · Answer 10 · 2026-04-14 03:03:44

127

Views

Number of elements in a finitely generated Dynkin system

Published on 14 Apr 2026 - 3:03

score 604 · Answer 11 · 2026-04-22 15:28:14

604

Views

Are polynomials dense in $H^\infty$

Published on 22 Apr 2026 - 15:28

score 441 · Answer 12 · 2026-04-15 10:13:11

441

Views

Stochastic Processes - Bass exercise 1.6

Published on 15 Apr 2026 - 10:13

score 882 · Answer 13 · 2026-04-12 23:25:45

882

Views

Absolute continuity of probability measure wrt Lebesgue measure

Published on 12 Apr 2026 - 23:25

score 155 · Answer 14 · 2026-04-18 02:24:47

155

Views

Evaluate the integral $\int_0^\infty \frac{\sin(ax)}{x} \frac{\sin(bx)}{x}e^{-kx}dx$

Published on 18 Apr 2026 - 2:24

score 40 · Answer 15 · 2026-04-16 11:56:12

40

Views

Find out whether the two statements about a stochastic process are equivalent

Published on 16 Apr 2026 - 11:56

List Question

If $f_n \rightarrow f$ is a sequence of $L^p$ and $g_n \rightarrow g$ a bounded sequence of $L^{\infty}$ then $f_ng_n \rightarrow fg$ in $L^p$

Show that $\lim_{n\to \infty}\int_{0}^{1}g(x)f(nx)dx = \Big(\int_{0}^{1}g(x)dx\Big)\Big(\int_{0}^{1}f(x)dx\Big)$

Showing that if $K \in L_{loc}^{\infty}(\mathbb{R}^+)$ then $|K^{\star n}| \leq M(a)^n \frac{x^{n-1}}{(n-1)!}$ where $M(a)$ is the supremum on $[0,a]$

Can $\limsup = \liminf$ only almost surely?

Fubini's Theorem for Radon complex measures

Confusing of the last line of proof for Fatou's lemma

Existence of random Variable $X$ such that $\lim_{n\to\infty} \mathbb{E}[(X_n - X)^2] = 0$

Measure defined by integral is non-atomic

Is there a name for a bijection that is bi-measurable?

Number of elements in a finitely generated Dynkin system

Are polynomials dense in $H^\infty$

Stochastic Processes - Bass exercise 1.6

Absolute continuity of probability measure wrt Lebesgue measure

Evaluate the integral $\int_0^\infty \frac{\sin(ax)}{x} \frac{\sin(bx)}{x}e^{-kx}dx$

Find out whether the two statements about a stochastic process are equivalent

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