It appears that these notations are equivalent when referring to the measure with which a function f ( x ) is integrated with respect to. It seems to me that the expression ∫ f d P X is very clear once some measure theory is learned. In what contexts are the other notations useful or necessary?

Question

It appears that these notations are equivalent when referring to the measure with which a function   f  (  x  ) is integrated with respect to. It seems to me that the expression   ∫  f  d      P    X   is very clear once some measure theory is learned. In what contexts are the other notations useful or necessary?

toriannucz · Accepted Answer

The   μ  (  d  y  ) notation is useful when you integrate with respect to a measure κ(x,⋅) on Y that varies with a parameter x. E.g. conditional probability or something like  f  (  x  )  =      ∫          Y        g  (  y  )  κ  (  x  ,  d  y  )  .The   P  (  X  ∈  d  x  ) is helpful because it makes certain formulas valid for arbitrary random variables look similar to corresponding ones for discrete random variables. E.g.  E  (  X  )  =      ∫                  R              x  P  (  X  ∈  d  x  )  .

It appears that these notations are equivalent when referring to the measure with which a function

Answered question

Answer & Explanation

New Questions in Pre-Algebra