yelinna (yelinna) wrote in traveling_pics,

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This week’s assignment from the League: Top 1

This is a little nerdy joke I wanted to make since some time ago. Enjoy!

First, let me introduce to you all, the amazing Wau number (I watched the video and I was amazed!):

After taking a course on Measure Theory the last months, I found another cool property of the Wau number. I won't get into the gory math details, don't get scared, this will be short and simple, let's begin:

We will define the following:

Omega is a space, any space: numbers, sets, animals, squares, Higgs bosons... anything you want.
F is a set called "sigma-algebra", it allow us to define properties on Omega and work with it. F has special characteristics: it includes a set that belongs to Omega and the complement of A. F includes Omega itself and the empty set. If F includes any other set, it includes its complement and the unions and intersections of any other set that belongs to F (also includes the complements of those unions and intersections).
Next we define a probability. It is just a number between 0 and 1 that we assign to any set in F.

These three together make a Measure Space (if P is any number it is called Measure Space. If P is between 0 and 1, it is a Probability Space. So any Probability Space is also a Measure Space):

Now I will introduce to you all the Lebesgue Integral. It is just a sum of the probabilities (or measures) of the sets inside F with other special properties, but we don't need to get into details here :)

Then, the whole family looks like this:

"dP" means that the integral is tied to our probability. A is any set in our sigma-algebra F.

This integral is the same as the following:

The "1A" means "The Indicator Function of set A". This is simply that if we tak any element in F, if it belongs to A, the indicator function gives 1, otherwise it gives 0.The Lebegue integral on set A is the same as the Lebesgue integral of the indicator function of A on the whole Omega space.

And this is equal to:

The probability of set A. The number between 0 and 1 we assigned to the set A.

If we tahe the Lebesgue Integral on the whole Omega space...

You can google to find out the real value of the Wau number :D you also will know why this is a "Top 1" :D :D

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