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 Post subject: Set of borel measurable functionsPosted: Fri, 18 May 2012 03:07:46 UTC
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Joined: Fri, 18 May 2012 03:00:09 UTC
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Consider collections of functions from to satisfying the following conditions:

(a) is a vector space

(b) contains the continuous functions

(c) If is an increasing sequence of nonnegative functions in and if exists and is finite for all , then .

Show that the collection consisting of the Borel-measurable functions is the smallest such collection of functions. (Hint: define . Show that contains the interval , and then contains the Borel sets).

characteristic function of .

if , if

I show that is in , but i try prove is a sigma-algebra but the union countable give me troubles, some help for it?

any help is apreciated, my first post

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 Post subject: Re: Set of borel measurable functionsPosted: Fri, 18 May 2012 05:07:26 UTC
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Joined: Mon, 29 Dec 2008 17:49:32 UTC
Posts: 6007
Location: 127.0.0.1, ::1 (avatar courtesy of UDN)
saiaN wrote:
Consider collections of functions from to satisfying the following conditions:

(a) is a vector space

(b) contains the continuous functions

(c) If is an increasing sequence of nonnegative functions in and if exists and is finite for all , then .

Show that the collection consisting of the Borel-measurable functions is the smallest such collection of functions. (Hint: define . Show that contains the interval , and then contains the Borel sets).

characteristic function of .

if , if

I show that is in , but i try prove is a sigma-algebra but the union countable give me troubles, some help for it?

any help is apreciated, my first post

.

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 Post subject: Re: Set of borel measurable functionsPosted: Sat, 19 May 2012 00:06:10 UTC
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Joined: Fri, 18 May 2012 03:00:09 UTC
Posts: 3
outermeasure wrote:

.

sorry but what is the arrow?, you could be more explicit please
thx

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 Post subject: Re: Set of borel measurable functionsPosted: Sat, 19 May 2012 00:09:20 UTC
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Joined: Wed, 30 Mar 2005 04:25:14 UTC
Posts: 12103
Location: Austin, TX
saiaN wrote:
outermeasure wrote:

.

sorry but what is the arrow?, you could be more explicit please
thx

monotone increasing.

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 Post subject: Re: Set of borel measurable functionsPosted: Sat, 19 May 2012 02:18:26 UTC
 S.O.S. Newbie

Joined: Fri, 18 May 2012 03:00:09 UTC
Posts: 3
great, just keep prove that is in
I will try to prove

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 Post subject: Re: Set of borel measurable functionsPosted: Sun, 20 May 2012 08:51:29 UTC
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Joined: Mon, 29 Dec 2008 17:49:32 UTC
Posts: 6007
Location: 127.0.0.1, ::1 (avatar courtesy of UDN)
saiaN wrote:
great, just keep prove that is in
I will try to prove

Have you seen the Dynkin's theorem (or lemma)?

Note that the exercise is a special case of the following version of the monotone class theorem:

Theorem Let be a collection of bounded functions closed under pointwise product, and let be the -algebra generated by . Let be a vector space (over ) of bounded real-valued functions on such that
(a) contains the constant functions; and
(b) if with and , then .
Then contains every bounded -measurable real-valued functions on .

The proof is essentially the same.

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