• Support PF! Buy your school textbooks, materials and every day products Here!

A caracterisation of f=0 by integrals

  • Thread starter quasar987
  • Start date
quasar987
Science Advisor
Homework Helper
Gold Member
4,771
7
[SOLVED] A caracterisation of f=0 by integrals

1. Homework Statement
Does anyone know how to show, or know a book that proves the implication

[tex]\left(\int_0^1f\varphi = 0 \ \ \forall \varphi \in C_c^1([0,1])\right)\Rightarrow f=0[/tex]

for f in L²([0,1]) and where [itex]C_c^1([0,1])[/itex] denotes the C^1([0,1]) functions whose support is contained in (0,1).

Thanks.


3. The Attempt at a Solution

I tried using the density of [itex]C_c^1([0,1])[/itex] in L^1 to obtained a sequence [itex]\varphi_n[/itex] that converges pointwise a.e. to the caracteristic function of [0,1] and then plugging-in the convergence theorems (Fatou, motone and dominated) but I eventually aknowledged that this would not work. Well, at least I got

[tex]\int_0^1f\leq 0[/tex]

out of Fatou.
 

Answers and Replies

morphism
Science Advisor
Homework Helper
2,013
4
Just to be clear, isn't [itex]C_c^1([0,1])[/itex] the set of compactly supported continuously differentiable funtions on [0,1]?

Anyway, here are some thoughts. Consider an open subset U of [0,1], whose characteristic function is [itex]\chi[/itex]. Get a sequence [itex]\{\varphi_n\}[/itex] in [itex]C_c^1([0,1])[/itex] that converges a.e. to [itex]\chi[/itex]. We can assume that [itex]M = \sup_n \| \varphi_n \|_\infty < \infty[/itex] (just construct them properly). So [itex]f\varphi_n \to f\chi[/itex] a.e., and [itex]\{f\varphi_n\}[/itex] is majorized by M|f|, which is integrable. Hence, by the dominated convergence theorem, [itex]\int_U f = 0[/itex] for all open subsets U of [0,1]. Now use the regularity of the Lebesgue measure to conclude that f=0 a.e.
 
quasar987
Science Advisor
Homework Helper
Gold Member
4,771
7
Just to be clear, isn't [itex]C_c^1([0,1])[/itex] the set of compactly supported continuously differentiable funtions on [0,1]?
Dunno if this is standard convention or not but my professor uses [itex]C_c^1([0,1])[/itex] to denote the set of compactly supported continuously differentiable funtions on (0,1). So for instance, the characteristic function of [0,1] is not in [itex]C_c^1([0,1])[/itex] according to my prof.

Anyway, here are some thoughts. Consider an open subset U of [0,1], whose characteristic function is [itex]\chi[/itex]. Get a sequence [itex]\{\varphi_n\}[/itex] in [itex]C_c^1([0,1])[/itex] that converges a.e. to [itex]\chi[/itex]. We can assume that [itex]M = \sup_n \| \varphi_n \|_\infty < \infty[/itex] (just construct them properly). So [itex]f\varphi_n \to f\chi[/itex] a.e., and [itex]\{f\varphi_n\}[/itex] is majorized by M|f|, which is integrable. Hence, by the dominated convergence theorem, [itex]\int_U f = 0[/itex] for all open subsets U of [0,1]. Now use the regularity of the Lebesgue measure to conclude that f=0 a.e.
Neat argument!

Say, have you seen my other similar question ?

https://www.physicsforums.com/showthread.php?t=226834
 
Last edited:

Related Threads for: A caracterisation of f=0 by integrals

Replies
1
Views
324
Replies
2
Views
2K
  • Last Post
Replies
8
Views
1K
Replies
5
Views
12K
  • Last Post
Replies
4
Views
1K
  • Last Post
Replies
3
Views
1K
Replies
1
Views
1K
Top