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riemann integrable
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mike lowry
science forum beginner


Joined: 20 May 2006
Posts: 7

PostPosted: Mon May 22, 2006 5:26 am    Post subject: riemann integrable Reply with quote
I have this problem,

Let A be a bounded subset of R, and suppose f: A -> R is a bounded Riemann integrable function. If f(x) > or = 0 for all x in A and the (intergral A) f = 0, prove the set A of 0 = { x in A : f(x) DNE 0 } has measure zero.

I know to start it by Defining A of n = {x in A : f(x) > 1/n}

however I am lost after that..

I appreciate any help, thanks.
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David C. Ullrich
science forum Guru


Joined: 28 Apr 2005
Posts: 2250

PostPosted: Mon May 22, 2006 11:14 am    Post subject: Re: riemann integrable Reply with quote
On Mon, 22 May 2006 01:26:07 EDT, mike lowry <mmehdiza@gmu.edu> wrote:

Quote:
I have this problem,

Let A be a bounded subset of R, and suppose f: A -> R is a bounded Riemann integrable function.
If f(x) > or = 0 for all x in A and the (intergral A) f = 0, prove the set A of 0 = { x in A : f(x) DNE 0 } has measure zero.

I know to start it by Defining A of n = {x in A : f(x) > 1/n}

Let's call that set A_n.

Quote:
however I am lost after that..

Show that A_n has measure zero, show that the set where f > 0 is
the union of the A_n, for n = 1, 2, ..., and prove or use the
previously-proved fact that a countable union of sets of measure
0 has measure 0.

Quote:
I appreciate any help, thanks.


************************

David C. Ullrich
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cody.roux@gmail.com
science forum beginner


Joined: 30 Apr 2006
Posts: 34

PostPosted: Mon May 22, 2006 2:23 pm    Post subject: Re: riemann integrable Reply with quote
David C. Ullrich a écrit :

Quote:
On Mon, 22 May 2006 01:26:07 EDT, mike lowry <mmehdiza@gmu.edu> wrote:

I have this problem,

Let A be a bounded subset of R, and suppose f: A -> R is a bounded Riemann integrable function.
If f(x) > or = 0 for all x in A and the (intergral A) f = 0, prove the set A of 0 = { x in A : f(x) DNE 0 } has measure zero.

I know to start it by Defining A of n = {x in A : f(x) > 1/n}

Let's call that set A_n.

however I am lost after that..

Show that A_n has measure zero, show that the set where f > 0 is
the union of the A_n, for n = 1, 2, ..., and prove or use the
previously-proved fact that a countable union of sets of measure
0 has measure 0.

I appreciate any help, thanks.


************************

David C. Ullrich


BTW the exact same demonstration works for positive Lebesgue-integrable
functions, a mutch more general case...
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David C. Ullrich
science forum Guru


Joined: 28 Apr 2005
Posts: 2250

PostPosted: Tue May 23, 2006 9:19 am    Post subject: Re: riemann integrable Reply with quote
On 22 May 2006 07:23:07 -0700, cody.roux@gmail.com wrote:

Quote:

David C. Ullrich a écrit :

On Mon, 22 May 2006 01:26:07 EDT, mike lowry <mmehdiza@gmu.edu> wrote:

I have this problem,

Let A be a bounded subset of R, and suppose f: A -> R is a bounded Riemann integrable function.
If f(x) > or = 0 for all x in A and the (intergral A) f = 0, prove the set A of 0 = { x in A : f(x) DNE 0 } has measure zero.

I know to start it by Defining A of n = {x in A : f(x) > 1/n}

Let's call that set A_n.

however I am lost after that..

Show that A_n has measure zero, show that the set where f > 0 is
the union of the A_n, for n = 1, 2, ..., and prove or use the
previously-proved fact that a countable union of sets of measure
0 has measure 0.

I appreciate any help, thanks.


************************

David C. Ullrich


BTW the exact same demonstration works for positive Lebesgue-integrable
functions, a mutch more general case...

The outline above looks the same, yes. The details, in particular
the details in showing that A_n has measure zero, would be a
little different (simpler in the case of Lebesgue measure,
actually.)


************************

David C. Ullrich
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