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fix: proofread Measure Theory part - #326

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fix: proofread Measure Theory part#326
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Proofreading pass over the Measure Theory part (issue #315), covering tex/measure/{measure-space,caratheodory,lebesgue-int,swapsum,pontryagin}.tex. All findings are minor: typos, grammar, and notation slips.

measure-space.tex

  • Power-set example: "if $\Omega$ is countably finite" → "countable" (the contrast in the same sentence is with $\Omega$ being "huge"/uncountable; "countably finite" is not a standard term).
  • "and in that case ware fine" → "we're fine".

caratheodory.tex

  • Compatibility summary: "$\mu\cme$ agrees with $\mu$" → "$\mu_0$" (at that point in the text only the pre-measure $\mu_0$ exists; the row-4 measure $\mu$ is introduced in the next section). Also "results ... means" → "mean".
  • "the subset $[0,1]$" → "the subset of $[0,1]$" (Cantor set).
  • "than a somewhat different Carathéodory theorem" → "then".
  • Proof of the pre-measure extension theorem: $\SA^\ast$ (undefined) → $\SA\cme$, the notation used everywhere else for the $\sigma$-algebra of $\mu^\ast$-measurable sets (two occurrences).

lebesgue-int.tex

  • "the Borel sigma-algebra" → "Borel $\sigma$-algebra" (consistency with the rest of the text).
  • "it is a good enough for us" → "it is good enough for us".
  • "(Monotoncity)" → "(Monotonicity)".

swapsum.tex

  • Pointwise-limit definition: stray comma in the sequence $f_3(\omega), \;, \dots$$f_3(\omega), \; \dots$.
  • Egorov's theorem: the sequence "converges to $f_U$" → "$f|_U$" (matches the restricted sequence $f_1|_U, f_2|_U, \dots$ just above).
  • Subject–verb agreement fixes: "we means" → "mean"; "$\int\dots$ and $\int\dots$ exists" → "exist"; "the convergence theorems ... is an attempt" → "are"; "The three big-name results ... is:" → "are:"; "no extra hypothesis" → "hypotheses"; "all the big-name theorem follows" → "theorems follow".

pontryagin.tex

  • Definition of topological group: doubled verb "A group $G$ is a \vocab{topological group} is a Hausdorff ..." → "A \vocab{topological group} $G$ is a Hausdorff ...".
  • "the Haar measure on a LCA group" → "on an LCA group".
  • $G = \ZZ/n\ZZ$ bullet labelled "discrete-time Fourier transform" but linking to the Discrete Fourier Transform article (and labelled "Discrete Fourier transform" in the summary table) → corrected to "discrete Fourier transform"; the discrete-time Fourier transform is the $G = \ZZ$ case, stated correctly elsewhere.
  • "Letting $G = Z$ for a finite group $G$" → "for a finite group $Z$".

No large mathematical errors were found in this part.

🤖 Generated with Claude Code


Generated by Claude Code

Proofreading pass over the Measure Theory part (issue #315):
typos, grammar, and notation fixes across measure-space,
caratheodory, lebesgue-int, swapsum, and pontryagin.

Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_011vpJKXaRH8HQbTeTqMwocZ
@vEnhance vEnhance added the proofreading-campaign Systematic full proofreading of text label Jun 29, 2026 — with Claude
@vEnhance vEnhance mentioned this pull request Jun 29, 2026
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vEnhance merged commit f097cd7 into main Jul 9, 2026
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