- #1

pswongaa

- 7

- 0

if lebesgue integral of f^2 over an interval equal 0, must f=0 a.e on that interval?

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- Thread starter pswongaa
- Start date

- #1

pswongaa

- 7

- 0

if lebesgue integral of f^2 over an interval equal 0, must f=0 a.e on that interval?

- #2

R136a1

- 343

- 53

- #3

- 15,450

- 687

What negative parts, R136a1? He's integrating f(x)

Is f a function that maps reals to reals, or something else?if lebesgue integral of f^2 over an interval equal 0, must f=0 a.e on that interval?

- #4

R136a1

- 343

- 53

Oh god. Never mind my reply.

- #5

Axiomer

- 38

- 5

Specifically, since [itex]f^2=|f^2|[/itex], this gives [itex]f^2=0[/itex] a.e., and hence [itex]f=0[/itex] a.e.

- #6

- 15,450

- 687

- #7

Axiomer

- 38

- 5

- #8

Axiomer

- 38

- 5

proof:

Define [itex]A=\{x\in X: g(x)≠0\}[/itex]. For all naturals n, define [itex]A_n=\{x\in X: |g(x)|>\frac{1}{n}\}[/itex].

[itex]\frac{1}{n}μ(A_n)=∫\frac{1}{n}x_{A_n}dμ≤∫|g|dμ=0[/itex], so [itex]μ(A_n)=0[/itex] for all n.

Then [itex]μ(A)=μ(\bigcup _{n=1}^∞A_n)≤\sum _{n=1}^∞μ(A_n)=0\implies μ(A)=0[/itex], as desired.

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