Notation ω:ε(w)=x in Koralov and Sinai's probability book

[adarpodracir]

Dear Physics Forums Community,

I am reading "Theory of Probability and Random Process" by Koralov L.B. and Sinai Y.G. and am having a hard time trying to understand the notation shown below (page 10).

In particular, what does
ω:ε(w)=x
mean below the second summation?

And how the authors could convert the single summation into that double one?

In this link, there is an image showing the notation employed by the authors:
i47.tinypic.com/jh8piq.png

As I am a new user, I cannot post links yet, so it will be greatly appreciated if you can check it manually.

Thank you very much for your time and consideration, and for any comments you could give me.

Ricardo.

 

[tiny-tim]

Welcome to PF!

Hi Ricardo! Welcome to PF!

In particular, what does
ω:ε(w)=x
mean below the second summation?

And how the authors could convert the single summation into that double one?

the colon (:) means "such that" …

so ω:ε(ω)=x means all ω such that ε(ω)=x​

the first step is partitioning Ω into classes with the same value of ε(ω) …

"all ω in Ω" is the same as "all ω in Ω such that ε(ω) = x, for all x in X"

 

[adarpodracir]

Dear tiny-tim,

Thank you very much for taking the time to reply.

Everything is clear now, but

Is there a theorem which says that:
"all ω in Ω" is the same as "all ω in Ω such that ε(ω) = x, for all x in X"

I did three examples, and they work perfectly nice! However, I would like to know if there is a theorem that supports it or if it follows from an intuitive idea.

Thank you!

 

[tiny-tim]

I would like to know if there is a theorem that supports it or if it follows from an intuitive idea.

Thank you!

obvious and intuitive really

let's see …

the theorem would begin "if ε is a function from Ω to X …"

how would you continue it? ​

 

[adarpodracir]

I think you're right.

Sometimes the most obvious and intuitive ideas/concepts in mathematics come with demonstrations. :)

Ricardo.