### Problem formulation

We have a probability density , with respect to a measure on .

If is mapped onto another set by a function

what will be the probability density over , with respect to a measure on ?

### General solution

where is Dirac’s delta function at so that

for any measurable subset of — in effect a conditional probability density with respect to the measure . If is a vector space, can be written .

**Proof**

We need prove that and implies that for any measurable subset of .

By definition, we have that | , | and | . |

From this, we can calculate

QED.

Â© 2008 Emanuel Winterfors code can be used in comments: $latex p(\theta)$ gives |

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The change of order of integration in the proof poses some restrictions on the integrand , in order to be valid. I’m not sure what the minimal conditions are, though…

Comment by winterfors — February 22, 2008 @ 11:36 am

This is just a change of variables in Lebesgue integral?

Comment by killua — March 27, 2008 @ 10:22 am

Not necessarily – the probability density can be defined in relation to any measure on , not necessarily the Lebesgue measure (even if it is certainly the most commonly used one).

Having looked into the problem a bit, I think the Fubini’s theorem should provide a guarantee for change of order of integration to be valid, as long as the integral of the absolute value of the integrand with respect to the Cartesian product measure exists. The existence and uniqueness of is in turn guaranteed by the Hahn-Kolmogorov theorem.

Comment by Emanuel Winterfors — March 27, 2008 @ 2:00 pm