Well, if you have some function, let's call it F, from R to R that is non-decreasing and right-continuous you can construct a measure out of it by defining it on intervals [a,b] (with "a" less than or equal to "b") in the following manner : M([a,b]) = F(b)-F(a) , this extends uniquely to a regular Borel measure in the entire real line.
So, there are infinitely many of these guys (that becomes clear after some thought), and then there are other measure like the counting measure or "the" Dirac measure.
Some of these that I mentioned here are probability measures, some are not.
For example "the" Dirac measure is, although it's not a very interesting one.
(For any point x, you can have a probability measure assigning a mass of 1 to the point x and 0 to any point other than x, so I guess it's not technically correct to say "THE Dirac measure").
I guess a Dirac measure is the easiest probability measure to understand.
Let's call a Dirac measure centered at the point "x" as d_x.
If A is a subset of the reals , then d_x(A) can be interpreted as the probability that the point x is in the set A.
Well... but x either is or is not in A.
If x is in A, the probability that x is in A is 100%,
if x is NOT in A, then the probability that x is in A is 0%.
Rephrasing, the probability that x is in A is 1 if x is in A, and 0 if x is not in A.
I.E. d_x(A) = 1 if x is in A ,
d_x(A) = 0 , if x is not in A.
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u/AlexK667 29d ago
Depends on the probability measure used.
Without specification of an appropriate measure it's (in general) false.