Interesting proof for the volume of an N-dimensional sphere
by greenturtle3141, Nov 9, 2021, 4:24 PM
Reading Difficulty: 5/5
Let
be the unit ball in 
. What is its Lebesgue measure, 
?
The answer somehow involves radicals of
, so "inspired" by this we start with the Gaussian integral:
We need to get to 
dimensions, so, well, just do it lol
To properly get into , we apply Fubini's Theorem (integrands are non-negative and clearly integrable) and induction on the RHS to turn this into an integral on the product space:
Now we apply , to get that this is equal to:
Hence 
.
Let
be the unit ball in 
. What is its Lebesgue measure, 
?The answer somehow involves radicals of
, so "inspired" by this we start with the Gaussian integral:
We need to get to 
dimensions, so, well, just do it lol
To properly get into , we apply Fubini's Theorem (integrands are non-negative and clearly integrable) and induction on the RHS to turn this into an integral on the product space:
Now we apply , to get that this is equal to:
and a measurable 
we have that:
This is not hard to show using the definition of Lebesgue integral. But if it makes you happy, we can be slick and write:![$$\int_0^\infty \mu\{x : f(x) > t\}\,dt = \int_0^\infty \int_{X}1_{f^{-1}((t,\infty))}\,d\mu\,dt = \int_X \int_0^\infty 1_{[0,f(x)]}\,dt\,d\mu = \int_X f(x)\,d\mu$$](http://latex.artofproblemsolving.com/7/0/f/70fb5dbc74a204c01c8004071c1b3d9c3d6ecf49.png)
Note that this is very similar to the identity 
for a continuous non-negative random variable 
.
Hence 
.This post has been edited 1 time. Last edited by greenturtle3141, Nov 9, 2021, 4:27 PM
March 2025
November 2024