How is √2 μ Interpreted Geometrically?

[Juliayaho]

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]

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]

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]

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?

 

[blue_raver22]

Geometrically, the measure √2 μ represents the length of the diagonal of a square with side length μ. This can be visualized as the distance from one corner of the square to the opposite corner, forming a right triangle with sides μ and μ. The measure √2 μ can also be interpreted as the hypotenuse of this right triangle. This is a fundamental concept in geometry and is often used to find the length of diagonal lines in various shapes. In real analysis, this measure can be used to calculate the distance between two points on the coordinate plane or to determine the length of curves in calculus.