I dont understand the following notation- could it be explained? $f_n$:D$\rightarrow$$\mathbb{R}$
Can someone just explain what the above notation means in literal words. The way i see it is that a function $f_n$ has a domain D which the function sends to a range of the real numbers?
Could someone also define the following:
$\mathbb{P}$:$\mathscr{B}$$\rightarrow$[0,1]
$A\rightarrow$$\mathbb{P}$(A)
To me this means that the probability that a function in the Borel set fit between 0 to 1 is equal to the probability of A going to P(A)?
 A: The notation $f_n : D \to \mathbb R$ means that you have a family of functions $f_1, f_2, f_3,\ldots$ which all take members of the set $D$ as inputs and give members of the set $\mathbb R$ as outputs.
If $D = \mathbb{R}$ then an example might be $f_n(x) = \frac{1}{n}\sin(nx)$. So, for example $f_1(x) = \sin x$ and $f_2(x) = \frac{1}{2}\sin 2x$, etc. You might like to ask yourself: what is the "limit function"? What is $f_{\infty}$?
If you have an $x$ in $D$ and it is sent by, say, $f_3$ to $y$ in $\mathbb {R}$ then you could write $f_3 : x \mapsto y$.
In your notation, you could say that $P : \mathscr{B} \to [0,1]$ and that $P : A \mapsto P(A)$.
Notice the difference between \to and \mapsto which give $\to$ and $\mapsto$. The latter applies to individual elements, e.g. $f: [0,1] \to [0,1]$, where $f : x \mapsto x^2$.
A: You are right. You have $$\underbrace{f}_\text{name of function} : \underbrace{D}_\text{domain} \to \underbrace{R}_\text{codomain} $$ with 


*

*domain = set of all arguments

*codomain = set of all possible values the function may attain (note, that not all values from the codomain must be attained)


So
$$\underbrace{\mathbb{P}}_\text{name of function}: \underbrace{\mathscr{B}}_\text{domain} \rightarrow \underbrace{[0,1]}_\text{codomain} : \underbrace{A}_\text{argument}\rightarrow\underbrace{\mathbb{P}(A)}_\text{value for argument A}$$
