Question

If $X$ is a Borel set and $f$ has countably many discontinuities, prove that $f:X\rightarrow \mathbf{R}$ is Borel measurable.

score 1.5k · Answer 1 · 2026-04-16 14:27:37

1.5k

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If $X$ is a Borel set and $f$ has countably many discontinuities, prove that $f:X\rightarrow \mathbf{R}$ is Borel measurable.

Published on 16 Apr 2026 - 14:27

score 595 · Answer 2 · 2026-04-17 02:32:15

595

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$m^(A \cup B) = m^(A) + m^*(B) $ implies $A$ and $B$ are measurable?

Published on 17 Apr 2026 - 2:32

score 37 · Answer 3 · 2026-04-15 01:04:22

37

Views

Determining whether an interval (0,1] $\in\mathfrak B(\mathbb R)$ or not.

Published on 15 Apr 2026 - 1:04

score 512 · Answer 4 · 2026-04-09 06:49:06

512

Views

Examples about bounded variation function

Published on 09 Apr 2026 - 6:49

score 500 · Answer 5 · 2026-04-13 00:22:46

500

Views

Problem in determining a map, given in the polar coordinate, is Area preserving or not on $\Bbb{R^2}$

Published on 13 Apr 2026 - 0:22

score 157 · Answer 6 · 2026-04-15 16:39:13

157

Views

Proving adaptedness of $ (\sum_{0 < s \leq t} \Delta X_s 1_{|\Delta X_s| > 1})_{0 \leq t \leq T}$ for RCLL adapted process $(X_t)_{0 \leq t\leq T}$.

Published on 15 Apr 2026 - 16:39

score 99 · Answer 7 · 2026-04-12 18:38:32

99

Views

Terminology question: $\sigma$-algebra generated on the codomain

Published on 12 Apr 2026 - 18:38

score 583 · Answer 8 · 2026-04-14 02:47:12

583

Views

Royden "Real Analysis", 3rd edition, chapter 4, exercise 9

Published on 14 Apr 2026 - 2:47

score 67 · Answer 9 · 2026-04-11 16:03:31

67

Views

Verification of result

Published on 11 Apr 2026 - 16:03

score 360 · Answer 10 · 2026-04-15 09:41:45

360

Views

Let $\Omega $ countable. Why the Borel $\sigma$-algebra is the power set?

Published on 15 Apr 2026 - 9:41

score 1.6k · Answer 11 · 2026-04-10 17:53:20

1.6k

Views

Fatou's Lemma proof from Royden 4th

Published on 10 Apr 2026 - 17:53

score 453 · Answer 12 · 2026-04-10 16:55:08

453

Views

What does mod 0 mean in the context of ergodic theory?

Published on 10 Apr 2026 - 16:55

score 893 · Answer 13 · 2026-04-17 01:16:42

893

Views

Problems proving that if $f_n\rightarrow f$ pointwise and $\int_R f=\lim_{n}\int_R f_n$ then $\int_E f=\lim_{n}\int_E f_n$ for meas $E \subseteq R$.

Published on 17 Apr 2026 - 1:16

score 1.2k · Answer 14 · 2026-04-16 07:25:59

1.2k

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If for each $\epsilon > 0$, there is an open set $O$ containing $E$ for which $m^*(O-E) < \epsilon$, then $E$ is measurable set.

Published on 16 Apr 2026 - 7:25

score 74 · Answer 15 · 2026-03-25 09:24:30

74

Views

Does Grothendieck's theorem hold for Bounded borel functions?

Published on 25 Mar 2026 - 9:24

List Question

If $X$ is a Borel set and $f$ has countably many discontinuities, prove that $f:X\rightarrow \mathbf{R}$ is Borel measurable.

$m^(A \cup B) = m^(A) + m^*(B) $ implies $A$ and $B$ are measurable?

Determining whether an interval (0,1] $\in\mathfrak B(\mathbb R)$ or not.

Examples about bounded variation function

Problem in determining a map, given in the polar coordinate, is Area preserving or not on $\Bbb{R^2}$

Proving adaptedness of $ (\sum_{0 < s \leq t} \Delta X_s 1_{|\Delta X_s| > 1})_{0 \leq t \leq T}$ for RCLL adapted process $(X_t)_{0 \leq t\leq T}$.

Terminology question: $\sigma$-algebra generated on the codomain

Royden "Real Analysis", 3rd edition, chapter 4, exercise 9

Verification of result

Let $\Omega $ countable. Why the Borel $\sigma$-algebra is the power set?

Fatou's Lemma proof from Royden 4th

What does mod 0 mean in the context of ergodic theory?

Problems proving that if $f_n\rightarrow f$ pointwise and $\int_R f=\lim_{n}\int_R f_n$ then $\int_E f=\lim_{n}\int_E f_n$ for meas $E \subseteq R$.

If for each $\epsilon > 0$, there is an open set $O$ containing $E$ for which $m^*(O-E) < \epsilon$, then $E$ is measurable set.

Does Grothendieck's theorem hold for Bounded borel functions?

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