questions about lebesgue measure

**PRLM** · December 21st 2009, 11:26 PM

$(X,F,\mu)=([-1,1],\Lambda,\lambda)$ , where $\Lambda$ is a sigma algebra of Lebesgue measurable sets and $\lambda$ is Lebesgue measure. Define $v(A)= \int_A x d\lambda$ for all $A \in \Lambda$ .

a) prove $v$ is a signed measure.
b) prove $v << \lambda$
c)Find the Hahn-Jordan decomposition of $v$ and its total variation.

i dont even know how to get start. please help me. and im not sure what << means on b).

**Shanks** · December 22nd 2009, 01:06 AM

1) varify that v satisfy the definition of sigh measure.
2) the v << u notation mean v is absolutely continous with respect to u, that is, if A is a measureable set with u(A)=0, then v(A)=0.
3)make sure that you understand the Hahn-Jordan decomposition and total variation of a measure in your textbook.

**PRLM** · January 1st 2010, 11:08 PM

Originally Posted by Shanks

1) varify that v satisfy the definition of sigh measure.
2) the v << u notation mean v is absolutely continous with respect to u, that is, if A is a measureable set with u(A)=0, then v(A)=0.
3)make sure that you understand the Hahn-Jordan decomposition and total variation of a measure in your textbook.

on 1), the definition says that v assumes at most one of $\infty$ , $-\infty$ . but how do i compute this integral? $\int_A x d\lambda \neq \int_A x dx$ , correct? how do i compute the smallest value and the biggest value it assumes? could you help me on this?

**Shanks** · January 2nd 2010, 12:01 PM

the equality holds since they define the same integral in the sense of lebesgue integration.
for the second problem,let f(x)=x, what is the positive part of f, and the negative part? the biggesr value it assumes is the integral of the positive part of f, similarly for the smallest value.

**PRLM** · January 2nd 2010, 12:24 PM

Originally Posted by PRLM

on 1), the definition says that v assumes at most one of $\infty$ , $-\infty$ . but how do i compute this integral? $\int_A x d\lambda \neq \int_A x dx$ , correct? how do i compute the smallest value and the biggest value it assumes? could you help me on this?

i am really not sure if this is correct so i would like someone to confirm what i did is correct, please.

1). to show that v is a signed measure, i verified that v checked the definition.
a)v assumes at most one of $\infty ,-\infty$ .
$vA=\int_A x d\lambda = \int_A x dx$ since $x$ is bounded on a finite interval and it is reimann integrable. and since the integral is all finite on any set $A \subset [-1,1]$ , v does not assume any of $\infty,-\infty$ .
b) $v(\emptyset)=0$ since the integral over a set of measure zero is zero.
c) by the equation $\int \Sigma f =\Sigma \int f$ , it is easy to verify.
Therefore it is a signed measure.

2).To show $|v|<<\mu$ , show that for each subset $A$ such that $\mu A=0$ , $|v|A=0$ . again since the integral over a set of measure zero is zero, it is obvious.

i am not sure what $\frac{dv}{d\mu}$ is. but i assume that it is Randon Nikodym derivative. Since v is a signed measure, by restricting v to a positive set, we get a measure. So v is restricted to [0,1]. So $\frac{dv}{d\mu}$ = x $\in [0,1]$ .

3)Hahn-Jordan decomposition is $v^+(E)=v(E \cap [0,1]),v^-(E)=-v(E \cup [-1,0))$ . $|v|(X)=\int_0^1 xdx-\int_{-1}^0 xdx = 1$ .

please someone correct me if im wrong.

**Shanks** · January 3rd 2010, 03:06 AM

the Radon Nikodym derivative is f(x)=x, x belongs to [-1,1].

**PRLM** · January 4th 2010, 01:27 AM

Originally Posted by Shanks

the Radon Nikodym derivative is f(x)=x, x belongs to [-1,1].

i thought that Radon N derivative is a nonnegative function.

**Shanks** · January 4th 2010, 03:19 AM

No, it is possible that the Radon Nikodym derivative assumes negative value.

**PRLM** · January 4th 2010, 01:27 PM

Originally Posted by Shanks

No, it is possible that the assumes negative value.

would you explain how u get Radon Nikodym derivative ?

**Shanks** · January 4th 2010, 08:54 PM

One way to do this is by the definition of derivative of a measure with respect to another measure.(it is complicated )
another easier way is using the Radon Nikodym theorem. it asserts the existence and uniqueness of Radon Nikodym derivative of a measure which is absolutely continous with respect to another measure.
for more information you can see Rudin's book Real and Complex Analysis.

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