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Questions regarding the Proof of Egorov's Theorem (Carothers)

score 517 · Answer 1 · 2026-04-13 09:56:03

Questions regarding the Proof of Egorov's Theorem (Carothers)

Published on 13 Apr 2026 - 9:56

score 101 · Answer 2 · 2026-04-11 19:21:42

Why are there different methods of constructing the outer measure leading up to Lebesgue measure (open covering or closed covering?)?

score 45 · Answer 3 · 2026-04-14 17:24:09

45

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Prove $m_*( \bigcup_{i=1}^\infty I_i ) = \sum_{i=1}^\infty \ell ( I_i )$ if $I_i \cap I_j = \emptyset \forall i,j$.

Published on 14 Apr 2026 - 17:24

score 6.7k · Answer 4 · 2026-04-16 20:42:51

6.7k

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Completion of borel sigma algebra with respect to Lebesgue measure

Published on 16 Apr 2026 - 20:42

score 12.1k · Answer 5 · 2026-04-10 20:47:29

12.1k

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Composition of measurable & continuous functions, is it measurable?

Published on 10 Apr 2026 - 20:47

score 229 · Answer 6 · 2026-04-17 12:51:08

229

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Simple question of the counting measure

Published on 17 Apr 2026 - 12:51

score 1.2k · Answer 7 · 2026-04-15 06:59:31

1.2k

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Measurable sets whose sum is not measurable.

Published on 15 Apr 2026 - 6:59

score 519 · Answer 8 · 2026-04-17 09:53:28

519

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$(f_n)$ Borel measurable implies $\sup_n f_n$ and $\inf_n f_n$ Borel measurable

Published on 17 Apr 2026 - 9:53

score 270 · Answer 9 · 2026-04-15 22:20:13

270

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$(f_n)$ Borel, $f(x)=\lim_{n\to\infty} f_n(x) \implies f$ Borel

Published on 15 Apr 2026 - 22:20

score 170 · Answer 10 · 2026-04-10 17:16:11

170

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Monotone Convergence Theorem in terms of infinite series

Published on 10 Apr 2026 - 17:16

score 405 · Answer 11 · 2026-04-15 06:12:34

405

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$f $ Borel measurable $\iff f^+, f^-$ are both Borel measurable

Published on 15 Apr 2026 - 6:12

score 115 · Answer 12 · 2026-04-11 23:25:37

115

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Lebesgue measurability of $(0,1)$

Published on 11 Apr 2026 - 23:25

score 6.8k · Answer 13 · 2026-04-17 08:05:23

6.8k

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Prove that if $E$ is measurable then $\forall \epsilon > 0$ $\exists F \subset E$ closed such that $m(E \setminus F) < \epsilon$.

Published on 17 Apr 2026 - 8:05

score 366 · Answer 14 · 2026-04-13 20:22:16

366

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Lebesgue Measure: show that a subset of R is equal to R

Published on 13 Apr 2026 - 20:22

score 40 · Answer 15 · 2026-04-15 05:19:57

40

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Prove $\exists x,y \in A$ st $\mu(A \cap (x,y))>c(x-y)$

Published on 15 Apr 2026 - 5:19

List Question

Questions regarding the Proof of Egorov's Theorem (Carothers)

Why are there different methods of constructing the outer measure leading up to Lebesgue measure (open covering or closed covering?)?

Prove $m_*( \bigcup_{i=1}^\infty I_i ) = \sum_{i=1}^\infty \ell ( I_i )$ if $I_i \cap I_j = \emptyset \forall i,j$.

Completion of borel sigma algebra with respect to Lebesgue measure

Composition of measurable & continuous functions, is it measurable?

Simple question of the counting measure

Measurable sets whose sum is not measurable.

$(f_n)$ Borel measurable implies $\sup_n f_n$ and $\inf_n f_n$ Borel measurable

$(f_n)$ Borel, $f(x)=\lim_{n\to\infty} f_n(x) \implies f$ Borel

Monotone Convergence Theorem in terms of infinite series

$f $ Borel measurable $\iff f^+, f^-$ are both Borel measurable

Lebesgue measurability of $(0,1)$

Prove that if $E$ is measurable then $\forall \epsilon > 0$ $\exists F \subset E$ closed such that $m(E \setminus F) < \epsilon$.

Lebesgue Measure: show that a subset of R is equal to R

Prove $\exists x,y \in A$ st $\mu(A \cap (x,y))>c(x-y)$

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