User contributions for RogierBrussee
A user with 1,476 edits. Account created on 26 July 2006.
19 July 2024
- 09:5409:54, 19 July 2024 diff hist +26 Monotone convergence theorem →Proof based on Fatou's lemma: lim = sup current
- 09:4709:47, 19 July 2024 diff hist +260 Monotone convergence theorem →Proof of theorem: Minor improvements.
- 09:1809:18, 19 July 2024 diff hist +37 Monotone convergence theorem →proof (lemma 3): make more explicit what \coprod means.
16 July 2024
- 16:4816:48, 16 July 2024 diff hist +23 Fatou–Lebesgue theorem →Proof: Kill limsup and liminf with one stone. current
- 16:3816:38, 16 July 2024 diff hist +358 Fatou–Lebesgue theorem →Proof: Make more clear where the finiteness of \int g d\mu comes in.
- 15:5815:58, 16 July 2024 diff hist +127 Fatou–Lebesgue theorem →Proof: The reverse Fatou lemma _is_ the Fatou lemma applied to g- f_n and cancelling.
15 July 2024
- 16:3216:32, 15 July 2024 diff hist −23 Monotone convergence theorem →Proof based on Fatou's lemma: Streamline the proof.
- 14:0414:04, 15 July 2024 diff hist +24 Monotone convergence theorem →proof (lemma 2): What we use is that a measure is countably addictive.
9 July 2024
- 15:5715:57, 9 July 2024 diff hist +4 Uniform integrability →Measure-theoretic definition: s/almost surely/almost everywhere/ current
- 05:4905:49, 9 July 2024 diff hist +117 Monotone convergence theorem →Proof of theorem: Clear up the lemma 1, 2, 3 mess.
- 05:3705:37, 9 July 2024 diff hist +2 Monotone convergence theorem →Intermediate results: I can count to three.
- 05:3605:36, 9 July 2024 diff hist +6 Monotone convergence theorem →Proof: Disentangle lemma 1, lemma 2 lemma 3 mess.
- 05:3205:32, 9 July 2024 diff hist +190 Monotone convergence theorem →Beppo Levi's lemma: Note the importance of the monotone convergence theorem.
- 05:2605:26, 9 July 2024 diff hist +8 Monotone convergence theorem →Convergence of a monotone series: Lemma 1 and 2 --> Proposition above.
- 05:2505:25, 9 July 2024 diff hist +195 Monotone convergence theorem →Convergence of a monotone sequence of real numbers: There are way to many lemma 1 and lemma 2 in this article
5 July 2024
- 14:2714:27, 5 July 2024 diff hist +116 Monotone convergence theorem →Proof based on Fatou's lemma
30 June 2024
- 12:5712:57, 30 June 2024 diff hist +113 Monotone convergence theorem →Proof of theorem: reorganise the argument a bit.
28 June 2024
- 15:3815:38, 28 June 2024 diff hist 0 Monotone convergence theorem →Convergence of a monotone series
25 June 2024
- 15:5215:52, 25 June 2024 diff hist +24 Monotone convergence theorem →Lebesgue integral as measure: more polish.
- 14:3014:30, 25 June 2024 diff hist +17 Monotone convergence theorem →Lebesgue integral as measure
- 14:2614:26, 25 June 2024 diff hist −222 Monotone convergence theorem →Lebesgue integral as measure: Improve the formulation of the lemma making it more correct and make the proof much easier again
21 June 2024
- 13:3613:36, 21 June 2024 diff hist +6 Monotone convergence theorem s/exchanged/interchanged
- 13:3413:34, 21 June 2024 diff hist +220 Monotone convergence theorem Note that the measure theoretic version is "the" monotone convergence theorem in more advanced math.
20 June 2024
- 16:4916:49, 20 June 2024 diff hist +336 Dominated convergence theorem →Statement: The caveat about limits and bounds being valid except on a set of measure 0 is needlessly alarming.
- 15:4915:49, 20 June 2024 diff hist +539 Dominated convergence theorem Note that the dominated convergence theorem says if |f_n| \le integrable then \lim \int = \int \lim
- 15:2015:20, 20 June 2024 diff hist +322 Dominated convergence theorem →Statement: Make the statement correct (we need to bound the whole sequence not just the point wise limit) and slightly easier to read.
19 June 2024
- 05:5905:59, 19 June 2024 diff hist −575 Monotone convergence theorem →Theorem (monotone convergence of non negative sums): The proof was more complicated than need be. Cheat a little by using that sup and sums can be interchanged for finite sums.
18 June 2024
- 21:1521:15, 18 June 2024 diff hist +34 Monotone convergence theorem →Theorem (monotone convergence of non negative sums)
- 21:1021:10, 18 June 2024 diff hist −13 Monotone convergence theorem →Theorem (monotone convergence of non negative sums): \sum_{I =1}^\infty is ugly.
- 21:0721:07, 18 June 2024 diff hist +1,009 Monotone convergence theorem →Theorem (monotone convergence of non negative sums): give a direct proof
- 20:1820:18, 18 June 2024 diff hist −111 Monotone convergence theorem →Lebesgue integral as measure: Make lemma 2 a bit lighter to the touch.
17 June 2024
- 09:0309:03, 17 June 2024 diff hist +13 Monotone convergence theorem →Proof: Another lim_{n \to \infty}
- 09:0109:01, 17 June 2024 diff hist +52 Monotone convergence theorem →Convergence of a monotone sequence of real numbers
- 08:5808:58, 17 June 2024 diff hist +19 Monotone convergence theorem →Theorem (monotone convergence theorem for non negative measurable functions): sups don really tend to infinity lims, lim sups and lim infs do.
- 08:5108:51, 17 June 2024 diff hist −760 Monotone convergence theorem →Intermediate results: Make the intermediate results less top heavy
- 08:0708:07, 17 June 2024 diff hist +269 Monotone convergence theorem →Proof of theorem: Explain the need for an epsilon of room.
- 07:4607:46, 17 June 2024 diff hist +4 Monotone convergence theorem →Proof of theorem: in step 3a.2 use implication instead of conjunction
16 June 2024
- 17:3517:35, 16 June 2024 diff hist −257 Monotone convergence theorem →Proof of theorem
- 17:1717:17, 16 June 2024 diff hist −806 Monotone convergence theorem →Proof of theorem: Step 3 is not so difficult either.
- 16:5616:56, 16 June 2024 diff hist −79 Monotone convergence theorem →Proof of theorem: step 2 really is trivial.
- 13:1313:13, 16 June 2024 diff hist +13 Fatou's lemma →Standard statement: make the main conclusion of the theorem display style and note that the lim inf may infinite.
- 09:4709:47, 16 June 2024 diff hist +51 Fatou's lemma →Proof: simplify the wording a bit and make the use of the monotone convergence theorem the main proof. It it is the natural thing to do.
15 June 2024
- 22:1122:11, 15 June 2024 diff hist −25 Fatou's lemma →Via the Monotone Convergence Theorem: change lims to sups
- 22:0122:01, 15 June 2024 diff hist +23 Fatou's lemma →Proof: lim g_n = sup g_n
- 21:5321:53, 15 June 2024 diff hist +12 Fatou's lemma →Standard statement: use \bar\R for extended reals
- 21:2221:22, 15 June 2024 diff hist +268 Monotone convergence theorem →Proof of theorem: switch t to 1-\varepsilon. More typing but also more suggestive
- 20:4020:40, 15 June 2024 diff hist −285 Monotone convergence theorem →Proof of theorem: Simplify step 1 a bit
- 17:3317:33, 15 June 2024 diff hist −155 Monotone convergence theorem →Proof of theorem: Move Fatou's lemma to the very end.
- 17:1917:19, 15 June 2024 diff hist −46 Monotone convergence theorem →Theorem (monotone convergence of non negative sums): the sup is clearer without the \infty
- 17:1517:15, 15 June 2024 diff hist +428 Monotone convergence theorem →Convergence of a monotone series: Explain relation of sum of non negative numbers with that of the sum of a series.
