# Meaning of probability measure

1. Mar 11, 2009

### David1234

What does it mean by
$$\int f(w) P (dw)$$

I don't really understand $$P (dw)$$ here. Does it mean $$P (x: x \in B(x, \delta))$$ for infinitely small $$\delta$$?

For example, with $$P(x)=1/10$$ for $$x=1, 2, ..., 10$$. How can we interpret this in term of the above integral

Thanks...

2. Mar 11, 2009

### John Creighto

I never took measure theory. Therefor my totally naive answer would be P(dw) is just a distribution function. Of course I'm probably completly wrong given I don't even know what a measure is.

3. Mar 11, 2009

### David1234

P(dw) is like a distribution fuction... may be. I am confused about P(dw), is it probability of dw? Then what is dw? Following the above example, say, we have $$f(w)=1$$ for w=1 and 0 otherwise. What is the meaning of dw here and hence value of P(dw) at w=1? I guess the above integral would give value = 1/10.

4. Mar 11, 2009

### HallsofIvy

Staff Emeritus
I have never seen "P(dw)". I think you mean what I would call dP(w)= P'(w)dw- the derivative of the cumulative probability distribution and so the probability density function. In that case, $\int F(w)dP= \int F(w)P'(w)dw$ is the expected value of F.

5. Mar 11, 2009

### David1234

I guess if P(w) has a derivative we can write it that way. I got the expression from a text book by Patrick Billingsley. Generally, when P(w) is not differentiable (as shown in the example), we can not write the expression in that form.

6. Mar 11, 2009

The notation
$$\int_\Omega X(\omega}) \, \mathcal{P}(dw)$$

is used in probability to indicate the expectation of the random variable $$X$$
with respect tot the probability measure (distribution) $$\mathcal{P}$$ over the probability space $$\Omega$$.

If $$\Lambda$$ is any measurable set, then

$$\int_\Lambda X(\omega) \, \mathcal{P}(dw) = E[X \cdot 1_{\Lambda}]$$

If the probability space is the real line with measure $$\mu$$, then

$$\int_\Lambda X(\omega) \, \mathcal{P}(dw) = \int_\Lambda f(x) \, \mu(dx)$$

is the Lebesgue-Stieltjes integral of $$f$$ with respect to the
probability measure $$\mu$$.

In more traditional form, if $$F$$ is the distribution function of $$\mu$$, and $$\Lambda$$ is an interval $$(a,b)$$, then

$$\int_\Lambda X(\omega) \, \mathcal{P}(dw) = \int_\Lambda f(x) \, \mu(dx) = \int_{(a,b)} f(x) \, dF(x)$$

If the probability measure doesn't have any atoms, the final integral is just a Lebesgue integral. If there are atoms, you need to take care to specify the interval according to whether the endpoints are or are not included - e.g.

$$\int_{a+0}^{b+0} f(x) \,dF(x), \quad \int_{a-0}^{b-0} f(x) \, dF(x)$$

and so on.

Billingsley is one of the "classic" probability texts. Chang's "A Course in Probability Theory" is another - I studied from it many years ago, and have the second edition. His writing is a little terse, but there is a lot packed into his book.

7. Mar 11, 2009

### David1234

Thanks a lot for the detail answer. I guess by Chang you mean Kai Lai Chung... :)

8. Mar 11, 2009

Yes, I did mean Kai Lai Chung - I would give a general description of my typing ability, but the description wouldn't be "safe for work".

9. Mar 12, 2009

### Focus

This isn't really correct. P may not be differentiable. When you say $\int_B f(x) P(dx)$, you are referring to the Lebesgue integral of f with respect to P. It is the same as saying $\int_B f dP$. You are just telling where the arguments lie so there is no confusion. I have to disagree with statdad in that it is not the Stieltjes integral, it is just the plain old Lebesgue integral. For Stieltjes you want to take a distribution function F of P and then you work it out as $\int_B f(x) dF(x)=\int_B f(x) P(dx)$.

Billingsley is a nice textbook and also I would recommend Ash, Real Analysis and Probability.

10. Mar 12, 2009

### David1234

Thanks...

I will have a look at "Real Analysis and Probability" by Ash.