Talk:Monotone convergence theorem

Revert from 2007 back to 2005

The following edit:

(cur) (last) 07:26, 19 November 2005 TakuyaMurata (Talk | contribs) (turn this into a disambig page) (undo)

(see [1]) replaced this page with a redirect page which did not preserve all the information. The fact is that the main theorem known as "monotone convergence" is Lebesgue's, and the article on Lebesgue's dominated convergence theorem does not discuss the monotone convergence theorem.

Loisel 05:43, 14 August 2007 (UTC)

Fixed some errors

I found some errors in the formulation of the monotone convergence theorem, so I fixed them and added a reference. I'm not sure though if the formulation I have is also the original formulation as given by Lebesgue. Perhaps someone could check and add a reference. I'm not sure where it was first published.
Mtroffaes (talk) 12:30, 22 February 2008 (UTC)

It was not an error. Since f_k(x) is monotone for every x, it converges to f(x), for every x. Loisel (talk) 18:50, 22 February 2008 (UTC)

My concern was with the integrability of the limit function: it could be that even though all f_k are measurable, f might not be. However, now looking at Schechter 21.3, and using the fact that [0,infty] can be turned into a metric space, it follows that f must be measurable too, so yes, the original formulation was therefore in fact correct. My guess is that Schechter adds the measurability condition on f because he only assumes convergence almost everywhere w.r.t. μ (which is weaker than pointwise convergence over the whole space). Anyway, the reference is wrong (21.38 states the convergence theorem for a.e. convergence, it does not state measurability of the limit!!!), I will fix it, and also update the theorem so it clearly applies to functions in the extended real space. Mtroffaes (talk) 11:04, 29 August 2012 (UTC)

Proof of the Lebesgue monotone convergence theorem

In the part of the proof concerned with the measurability of f it says:

On the other hand, since [0, t] is a closed interval,

$f(x)\in I\Leftrightarrow f_{k}(x)\in I,~\forall k\in \mathbb {N} .$

I haven't got a math textbook in front of me, but I think I remember from my math studies it should rather read something like:

On the other hand, since [0, t] is a closed interval,

$f(x)\in I\Leftrightarrow \exists ~N>0~|~f_{k}(x)\in I,~\forall k\geq N.$ —Preceding unsigned comment added by 212.126.224.100 (talk) 13:48, 5 October 2010 (UTC)

Not quite - I think you're getting confused with convergence of sequences. In fact, your two "right hand sides" of the equivalences are actually equivalent to one another. Indeed, if the first "right hand side" holds, then clearly the second one does (for $N=1$, say). If the second "right hand side" holds, then as $0\leq f_{k}(x)\leq f_{k+1}(x)$ for all k, the first "right hand side" must also hold.

So, your second statement is technically correct as well as the first, but it's overly complicated and somewhat misleading. Hope this helps. :-) Tcnuk (talk) 14:17, 5 October 2010 (UTC)

In the proof, why is ${\mathcal {B}}_{\mathbb {R} _{\geq 0}}$ defined as the Borel set generated by the nonnegative extended real numbers? This should probably be ${\mathcal {B}}_{{\overline {\mathbb {R} }}_{\geq 0}}$. Also, it seems rather artificial to restrict the Borel set to the nonnegative extended reals, rather than just using ${\mathcal {B}}_{\overline {\mathbb {R} }}$, but that's just my opinion. Hexidominus —Preceding undated comment added 10:07, 5 June 2019 (UTC)

Readability for casual readers

Not knowing what a monotone sequence is, this is an impossible article to read. Could someone please edit it so that it's clear what the theorems do?Bigben987 (talk) 21:00, 10 March 2012 (UTC)

I agree. I expanded the lead a bit to explain the intuitive idea behind the theorems. --Mark viking (talk) 20:45, 24 June 2013 (UTC)

Erroneous wording

The introduction contains the following: "... monotonic sequences (sequences that are increasing or decreasing) that are also bounded ...". In fact, the definition in the parentheses conflicts with the definition on the page to which "monotonic sequences" links.

Technically, a monotonic sequence is non-decreasing or non-increasing (see monotonic sequence and [2]). When the sequence is always increasing or always decreasing, then we call it "strictly monotonic". Examples:

 A_n = 1, 2, 3, 3, 3, 4, 5, 5, ... = monotonic
 B_n = 1, 2, 3, 4, 5, 6, ... = strictly monotonic
 C_n = 1, 1, 1, 1, ... = monotonic

- 216.30.180.227 (talk) 17:14, 27 February 2014 (UTC)

unncessary integral condition

in the lebesgue segment of the article, under Remark 5, it is said that f,g be function into the nonnegative real numbers and then there is the additional condition that the integral may differ from infinity minus infinity (i.e. exist). doesnt that follow from the nonnegativity of f and g already?

also, in the used form of the actual theorem, the sequence shall be measurable with respect to the Borel sigma algebra on the nonnegative real numbers, but is allowed to have the value infinity. how can it have value infinity if infinity is not in any of the borel sets?

On your first point, thank you very much! This was a definite oversight.

On the second one, $\operatorname {\mathcal {B}} _{\mathbb {R} _{\geq 0}}$ is the $\sigma$-algebra generated by Borel sets in $\mathbb {R} _{\geq 0}$ and $\{+\infty \}.$ StrokeOfMidnight (talk) 21:29, 17 January 2018 (UTC)