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[SOLVED] Interpret Measure Geometrically

Juliayaho

New member
Apr 11, 2013
13
This is part 2 of a question... I already solved part 1 but I can't seem to be able to solve this one.
Interpret the measure √2 μ geometrically?

Any ideas... This is from real analysis class

Thanks in advance!
 

Prove It

Well-known member
MHB Math Helper
Jan 26, 2012
1,404
Re: Measure.... Geometrically?

This is part 2 of a question... I already solved part 1 but I can't seem to be able to solve this one.
Interpret the measure √2 μ geometrically?

Any ideas... This is from real analysis class

Thanks in advance!
What is \(\displaystyle \displaystyle \mu \) representing?
 

Juliayaho

New member
Apr 11, 2013
13
Re: Measure.... Geometrically?

The unique positive regular measure on B(R^2) such that for all f in C_c(R^2) λf= ∫_R^2 f dµ

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What is \(\displaystyle \displaystyle \mu \) representing?
The unique positive regular measure on B(R^2) such that for all f in C_c(R^2) λf= ∫_R^2 f dµ

- - - Updated - - -

The unique positive regular measure on B(R^2) such that for all f in C_c(R^2) λf= ∫_R^2 f dµ

- - - Updated - - -



The unique positive regular measure on B(R^2) such that for all f in C_c(R^2) λf= ∫_R^2 f dµ
λf=∫(from R to Riemann) f(tt)dt for all f in C_c(R^2).
 

Ackbach

Indicium Physicus
Staff member
Jan 26, 2012
4,197
Re: Measure.... Geometrically?

I could be wrong, but it seems to me that this question could be answered much more easily (that is, intuitively) by thinking about the corresponding Riemann integral. It's a fact that if the function is Riemann integrable, then
$$\int_{ \mathbb{R}^{2}} f \, dA = \int_{ \mathbb{R}^{2}} f \, d\mu.$$
So take an example, and make $f$ really simple, say $f=1$. Then your integral is simply computing area. Your new measure, $\sqrt{2} \mu$, simply multiplies your area by $\sqrt{2}$, since constants pull out of integrals (at least, they pull out of Riemann and Lebesgue integrals). What is going on geometrically for that to happen?