Let \(F: \mathbb{R} ightarrow[0,1]\) be a distribution function.

a) Show that there exists a probability space \((\Omega, \mathscr{A}, \mathbb{P})\) and a random variable \(X\) such that \(F(x)=\mathbb{P}(X \leqslant x)\).

b) Show that there exists a probability space \((\Omega, \mathscr{A}, \mathbb{P})\) and an iid sequence of random variables \(X_{n}\) such that \(F(x)=\mathbb{P}\left(X_{n} \leqslant xight)\).

c) State and prove the corresponding assertions for the \(d\)-dimensional case.