“Measure Theory becomes
Probability Theory when Euclidean Space becomes Probability Space”
In probability theory we model
the probability of a complex real phenomenon that is random in nature. Probability
of occurrence of outcomes of such phenomenon is dependent on several variables
that are also random in nature. This can be explained by a multidimensional Euclidean
Space with one coordinate axes being the probability; this probability is a
function of these multiple random variables. So a multi-dimensional probability
space has to be mentally visualized. If
there is only one variable then the probability space becomes a two dimensional
Euclidean space
As we go closer and closer to predicting the complexity of
real life phenomenon by computing its probability, the simple formula of favorable
cases/total cases takes an entirely new form. This form is governed by axioms
of probability theory which can be closely linked to measure theory. For a two
dimensional probability space computation of probability for a continuous
random variable is equal to finding an area. This is done using simple integration,
as shown figure above. Please refer to blog of 26 Feb 2017 for the details of
these figures. For a three dimensional space it is a volume and we use double integrals
to find this volume. But for n dimensional space, it is Lebesgue measure
(measure theory) that aids in computation of the n dimensional volume. For ease
of graphical representation the figure below shows a three dimensional space.
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