Question

Show that $0\leq\|f\|^2+2 r\int_X |fg|d\mu+r^2\|g\|^2$ for all $r\in\mathbb{R}$ implies that $\int_X |fg|d\mu\leq\|f\|\|g\|$.

score 60 · Answer 1 · 2022-06-02 18:21:14

60

Views

Show that $0\leq\|f\|^2+2 r\int_X |fg|d\mu+r^2\|g\|^2$ for all $r\in\mathbb{R}$ implies that $\int_X |fg|d\mu\leq\|f\|\|g\|$.

Published on 02 Jun 2022 - 18:21

score 67 · Answer 2 · 2022-06-03 01:42:31

67

Views

Show $f, g\in\mathscr{L}^2(\mu)$ imply $f\overline{g}, \overline{f}g\in\mathscr{L}(\mu)$, and $\int f\overline{g} +\overline{f}g\leq2\|f\|\|g\|$.

Published on 03 Jun 2022 - 1:42

score 75 · Answer 3 · 2022-06-03 17:55:15

75

Views

Show "if $f\geq0, f\in\mathscr{L}^2$ then there is $\{g_n\}$ continuous such that $\|f-g_n\|\to0$" implies the same for all $f\in\mathscr{L}^2$.

Published on 03 Jun 2022 - 17:55

score 54 · Answer 4 · 2022-06-04 09:42:08

54

Views

Checking Conditional Probability on Discrete Random Variable

Published on 04 Jun 2022 - 9:42

score 125 · Answer 5 · 2022-06-05 13:30:14

125

Views

Fatou's theorem

Published on 05 Jun 2022 - 13:30

score 90 · Answer 6 · 2022-06-07 12:16:51

90

Views

$L_\infty$ norm and uniform convergence with Lebesgue integral

Published on 07 Jun 2022 - 12:16

score 164 · Answer 7 · 2022-06-08 20:45:31

164

Views

Theorem 1.8 of Papa Rudin

Published on 08 Jun 2022 - 20:45

score 31 · Answer 8 · 2022-06-09 16:33:00

31

Views

Measurability of functions - Is $f$ measurable?

Published on 09 Jun 2022 - 16:33

score 93 · Answer 9 · 2022-06-09 20:16:57

93

Views

Theorem 1.12(c) of Papa Rudin

Published on 09 Jun 2022 - 20:16

score 45 · Answer 10 · 2022-06-10 22:58:57

45

Views

proving a function with unit absolute value is measurable

Published on 10 Jun 2022 - 22:58

score 53 · Answer 11 · 2022-06-11 11:39:13

53

Views

step function integral and measurable

Published on 11 Jun 2022 - 11:39

score 68 · Answer 12 · 2022-06-12 01:52:32

68

Views

Show that $\sum_{k=1}^{\infty}\int_X|gf_k|d\mu<+\infty$ implies $|g|\sum_{k=1}^{\infty}|f_k|<+\infty$ almost everywhere on X.

Published on 12 Jun 2022 - 1:52

score 29 · Answer 13 · 2022-06-12 03:48:50

29

Views

Show that $\mu(a)=1$ for some $a$ if $\mu(f^{-1}([a_n, b_n)))=1$ and $\lim_{n\to\infty}(b_n-a_n)=0$.

Published on 12 Jun 2022 - 3:48

score 121 · Answer 14 · 2022-06-14 00:11:34

121

Views

Failure of the Lebesgue differentiation theorem

Published on 14 Jun 2022 - 0:11

score 77 · Answer 15 · 2022-06-22 13:43:58

77

Views

Why is the uniform limit of strongly measurable functions in turn strongly measurable?

Published on 22 Jun 2022 - 13:43

List Question

Show that $0\leq\|f\|^2+2 r\int_X |fg|d\mu+r^2\|g\|^2$ for all $r\in\mathbb{R}$ implies that $\int_X |fg|d\mu\leq\|f\|\|g\|$.

Show $f, g\in\mathscr{L}^2(\mu)$ imply $f\overline{g}, \overline{f}g\in\mathscr{L}(\mu)$, and $\int f\overline{g} +\overline{f}g\leq2\|f\|\|g\|$.

Show "if $f\geq0, f\in\mathscr{L}^2$ then there is $\{g_n\}$ continuous such that $\|f-g_n\|\to0$" implies the same for all $f\in\mathscr{L}^2$.

Checking Conditional Probability on Discrete Random Variable

Fatou's theorem

$L_\infty$ norm and uniform convergence with Lebesgue integral

Theorem 1.8 of Papa Rudin

Measurability of functions - Is $f$ measurable?

Theorem 1.12(c) of Papa Rudin

proving a function with unit absolute value is measurable

step function integral and measurable

Show that $\sum_{k=1}^{\infty}\int_X|gf_k|d\mu<+\infty$ implies $|g|\sum_{k=1}^{\infty}|f_k|<+\infty$ almost everywhere on X.

Show that $\mu(a)=1$ for some $a$ if $\mu(f^{-1}([a_n, b_n)))=1$ and $\lim_{n\to\infty}(b_n-a_n)=0$.

Failure of the Lebesgue differentiation theorem

Why is the uniform limit of strongly measurable functions in turn strongly measurable?

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