# Proper definition of complex random variable.

According to Wikipedia,

A complex random variable $$Z$$ on the probability space$$(\Omega,{\mathcal {F}},P)$$ is a function $$Z\colon \Omega \rightarrow \mathbb {C}$$ such that both its real part $$\Re {(Z)}$$ and its imaginary part $$\Im {(Z)}$$ are real random variables on $$(\Omega ,{\mathcal {F}},P)$$.

Can we define a complex random variable as follows:

A complex random variable $$Z$$ on the probability space$$(\Omega,{\mathcal {F}},P)$$ is a function $$Z\colon \Omega \rightarrow \mathbb {C}$$ such that $$\{\omega \in \Omega |Z(\omega )\in S\} \in \mathcal{F}$$ for all $$S\in\mathcal{G}$$ with $$\mathcal{G}$$ being the Borel $$\sigma$$-algebra on $$\mathbb{C}$$.

Are the above two definitions equivalent?

In general, a random variable is none other than a measurable function between two measure spaces (a measure space is a set equipped with a $$\sigma$$-algebra). Your second definition is a special case of this more general definition. To see why it is equivalent to the first definition, we need to understand the relationship between the Borel $$\sigma$$-algebras on $$\mathbb R$$ and $$\mathbb C$$.
We can use a nice property of the Borel $$\sigma$$-algebra on any topological space $$X$$: a function $$f\colon (\Omega,\mathcal F)\to (X,\mathcal B(X))$$ is measurable if and only if $$f^{-1}(U)$$ is measurable for all open sets $$U\subseteq X$$. Now since $$\mathbb C$$ is equipped with the product topology of $$\mathbb R\times \mathbb R$$, a set $$U$$ is measurable if and only if $$\pi_1(U)$$ and $$\pi_2(U)$$ are measurable, where $$\pi_i$$ denote the projections onto the coordinates. Thus, it follows that the first condition is equivalent to the second, as a result of the fact that $$\mathbb C$$ is equal to the product topology of two copies of $$\mathbb R$$.
The Borel sigma algebra of $$\mathbb C$$ is equal to the sigma algebra generated by sets of the form $$\{(x+iy: x \in A, y\in B\}$$ where $$A$$ and $$B$$ are Borel sets in $$\mathbb R$$. Hence $$Z$$ is measurable iff $$\Re Z^{-1}(A) \cap \Im Z^{-1}(B)\in \mathcal F$$ for such $$A$$ and $$B$$. So the two definitions are equivalent.