prove that every field contains empty set and the whole space omega
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In probability theory, a probability space or a probability triple {\displaystyle (\Omega ,{\mathcal {F}},P)}(\Omega ,{\mathcal {F}},P) is a mathematical construct that provides a formal model of a random process or "experiment". For example, one can define a probability space which models the throwing of a dice.
A probability space consists of three elements:[1][2]
A sample space, {\displaystyle \Omega }\Omega , which is the set of all possible outcomes.
An event space, which is a set of events {\displaystyle {\mathcal {F}}}{\mathcal {F}}, an event being a set of outcomes in the sample space.
A probability function, which assigns each event in the event space a probability, which is a number between 0 and 1.
In order to provide a sensible model of probability, these elements must satisfy a number of axioms, detailed in the article.
In the example of the throw of a standard die, we would take the sample space to be {\displaystyle \{1,2,3,4,5,6\}}{\displaystyle \{1,2,3,4,5,6\}}. For the event space, we could simply use the set of all subsets of the sample space, which would then contain simple events such as {\displaystyle \{5\}}{\displaystyle \{5\}} ("the die lands on 5"), as well as complex events such as {\displaystyle \{2,4,6\}}{\displaystyle \{2,4,6\}} ("the die lands on an even number"). Finally, for the probability function, we would map each event to the number of outcomes in that event divided by 6 — so for example, {\displaystyle \{5\}}{\displaystyle \{5\}} would be mapped to {\displaystyle 1/6}1/6, and {\displaystyle \{2,4,6\}}{\displaystyle \{2,4,6\}} would be mapped to {\displaystyle 3/6=1/2}{\displaystyle 3/6=1/2}.
When an experiment is conducted, we imagine that "nature" "selects" a single outcome, {\displaystyle \omega }\omega , from the sample space {\displaystyle \Omega }\Omega . All the events in the event space {\displaystyle {\mathcal {F}}}{\mathcal {F}} that contain the selected outcome {\displaystyle \omega }\omega are said to "have occurred". This "selection" happens in such a way that were the experiment repeated many times, the number of occurrences of each event, as a fraction of the total number of experiments, would tend towards the probability assigned to that event by the probability function {\displaystyle P}P.
The Russian mathematician Andrey Kolmogorov introduced the notion of probability space, together with other axioms of probability, in the 1930s. In modern probability theory there are a number of alternative approaches for axiomatization — for example, algebra of random variables.