I was wondering what the typical approach is for creating a "mathematical object" such as the probability model kolmogorov made (I've also heard it called a probability space...not really sure what the difference is)...(adsbygoogle = window.adsbygoogle || []).push({});

[itex] <\Omega,\mathcal{F},P> [/itex], where [itex] \Omega [/itex] is the sample space, [itex] \mathcal{F} [/itex] is the [itex] \sigma[/itex]-field over [itex] \Omega [/itex], and [itex] P [/itex] is the map [itex] P : \mathcal{F} \rightarrow [0,1] [/itex]. The idea being that 2 probability models are the same only if each of these 3 things are all identical...

Can you make groups like this for anything? e.g....

...I read in my text book that "Two functions are the same if and only if they have the same Domain, Codomain, and Rule mapping from the Domain to the Co-Domain." ....Does this mean that I could augment a given function defined unrigorously as y = f(x) as a similarly constructed object like, say...

[itex] <{\textbf{X}},{\textbf{Y}},f> [/itex], where [itex]{\textbf{X}} [/itex] is the domain, [itex] {\textbf{Y}} [/itex] is the co-domain, and [itex] f [/itex] maps from the domain to the co-domain .

I realize this is often a pointless model to make ( other than in probability which has complications in defining the domain )... you can say all this with just ... [itex] f: {\textbf{X}} \rightarrow {\textbf{Y}} [/itex] but aside from its unnecessary-ness. is there any reason I can't do this...

Where can I study this in more depth/ what's studying functions in this depth called? Any recommended Texts on this area?

Thank you All!!!!

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# Defining mathematical objects

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