Sample space (S) is a set of all possible outcomes of a random experiment.
$S = \{x | x $is an outcome$\}$
Event (E) is a subset of sample space.
$E \subseteq S$
Probability is a function from a set (Event) to $[0,1]$
$p : E -> [0,1]$
My doubt is that whether conditional probability is also a form of probability, whose argument is a set? Is conditional probability a function?
If yes, what is the argument of the conditional probability function?
For an event $E$ s.t. $P(E)>0$, the answer is yes. Since $P(A|E)=\frac{P(A\cap E)}{P(E)}$ , we can define a new probability measure $P_E$ s.t. $P_E(A)=P(A|E)=\frac{P(A\cap E)}{P(E)}$. All the properties of a probability measure hold for $P_E$.
In some cases it is possible to define a conditional probability measure on events $E$ s.t. $P(E)=0$, but that is more complicated.