In a probability space $(\Omega , \mathcal{F} , P)$, what is the meaning of $P(d w)$

Question

I am reading a book on reinforcement learning. The author says, let $(\Omega , \mathcal{F}, P)$ be a probability space. Let $(\mathbb{R}, \mathcal{B})$ be a measurable space, where $\mathcal{B}$ is the smallest sigma-algebra of subsets of $\mathbb{R}$ that contains all the open intervals in $\mathbb{R}$ (which means $\mathcal{B}$ is a Borel set). Then the author says, let $X$ be a measurable map from $(\Omega , \mathcal{F}, P)$ to $(\mathbb{R}, \mathcal{B})$ (which means $X$ is a random variable). Then he goes on to say that the expected value of $X$ is \begin{equation} E(X)=\int_{w\in\Omega} X(w)P(\mathrm{d}w). \end{equation}

Now the thing I don't understand is, what is $P(\mathrm{d}w)$? Since $P:\mathcal{F}\rightarrow [0,1]$, and the author is introducing $P(\mathrm{d}w)$, which implies $\mathrm{d}w\in \mathcal{F}$? Then please explain to me how $\mathrm{d}w\in \mathcal{F}$?

My intuition says $E(X)=\int_{w\in\Omega} X(w) P(w)$, not $\int_{w\in\Omega} X(w) P(\mathrm{d}w)$. I am attaching the screenshot of the book I am reading.

Answer 1

It is just notation. If $\mu$ is a measure on a measurable space $X$, and $f$ a (complex) measurable function on $X$, we have (by definition) $$\int_X f d\mu =\int_X f(x)d\mu(x)=\int_X f(x) \mu(dx).$$

Answer 2

What Quantum said. Perhaps an explanation for the notation $\int f(x)\;\mu(dx)$:

Approximate $f$ by a simple function; then the integral is approximated by a sum $$ \sum_{j=1}^n f(x_i)\mu(A_i) $$ where $\{A_1,\dots,A_n\}$ is a partition of $\Omega$ and $x_i \in A_i$. Now take a "limit", where the partition gets finer, the sets get smaller, and the result is symbolized as $$ \int_\Omega f(x)\;\mu(dx) $$ where we think of "$dx$" as a very small set containing $x$.

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Usefulness of the notation. Sometimes we have a "transition probabiliy" in the form $T(x,A)$, where for each fixed $x$, the measure $A \mapsto T(x,A)$ describes probability distribution for the new position one step later. Then we may want to integrate with respect to that measure, parametrized by $x$. So we want integrals of the form $$ \int \phi(x)\;T(x,dx) $$ to signify, for each $x$, integration with respect to the measure $A \mapsto T(x,A)$. I think it would be a bit confusing to write something like $$ \text{?}\quad\int\phi(x)\;dT(x,x) $$