Understanding the probability density function

Question

I know the definition of a probability density function and also used it several times. Now I am reading a proof in Jacod&Protter and they assume the following:

If the Random variable $Y$ has a density $f$, then $\mathbb{P}_Yd(y)=f_Y(y)dy$. This is vital to the proof and I would like to understand that. Isn't $$ \mathbb{P}(A):=\int_A f(x)dx $$ How do they get to this equality then? For a better understanding I will just copy the part where it is needed.

Answer 1

By definition of the density function, we have (where $\Omega$ denotes the set of events) \begin{equation}\tag{1}\label{1}\int_\Omega 1_A(x) \,\mathbb P_Y(\mathrm dx) = \mathbb P_Y(A)=\int_\Omega 1_A(x) f_Y(x)\,\mathrm dx,\end{equation} where $A\subset\Omega$ is any measurable set and $1_A$ its characteristic function.

Note that \eqref{1} also holds for any simple function.

Let $f$ be a positive, $\mathbb P_Y$-measurable, $\mathbb P_Y$-integrable function.
By a standard result from measure theory, we know that for any sequence of positive simple functions $f_n$ such that $f_n(x)\to f(x)$ for all $x\in\Bbb R$ and $f_1\le f_2\le\dots\le f_n\le\dots\le f$, we have $$\int_\Omega f\, \mathrm d\mathbb P_Y=\lim_{n\rightarrow\infty}\int_\Omega f_n\, \mathrm d\mathbb P_Y.$$

Fix any sequence $f_n$ as above (such a sequence exists if and only if $f$ is measurable). Then \begin{split}\int_\Omega f\, \mathrm d\mathbb P_Y&=\lim_{n\rightarrow\infty}\int_\Omega f_n\, \mathrm d\mathbb P_Y \\&\overset{f_n \text{ are simple functions}}=\lim_{n\to\infty} \int_\Omega f_n(x)f_Y(x)\,\mathrm dx \\& \overset{\text{Monotone convergence Thm.}}= \int_\Omega \lim_{n\to\infty} f_n(x)f_Y(x) \,\mathrm dx \\& = \int_\Omega f(x) f_Y(x) \,\mathrm dx. \end{split}

From the definition of the integral, we can conclude that for all $\mathbb P_Y$-measurable, integrable functions $f$, we have

$$\int_\Omega f(x) \,\mathbb P_Y(\mathrm dx)=\int_\Omega f(x) f_Y(x)\,\mathrm dx.$$

This is what they meant to say while writing (the somewhat confusing) „$\mathbb P_Y(\mathrm dx)=f_Y(x)\mathrm dx$“.

(Short side note: $\mathrm dx$ of course denotes integration w.r.t. Lebesgue measure.)