In probability theory, the notion of independence of random variables is usually introduced as follows: we say that two random variables $X, Y$ are independent if for any two real numbers $(x,y)$ one has: $\mathbf{P} \{ X \le x , Y \le y \} = \mathbf{P} \{ X \le x \} \cdot \mathbf{P} \{ Y \le y \}$ (i.e, the CDF of the pair $(X,Y)$ is just the product of the CDFs of $X$ and $Y$).
However, if we go back to the formal definition of random variables as measurable functions, then independence basically means that the sigma algebras $\sigma (X)$ and $\sigma (Y)$ generated by $X, Y$ (i.e., $\sigma (A)$ is the smallest sigma algebra inside which $A$ is measurable) are independent. This definition allows us to speak about independence of more general class of random variables, than just $\mathbf{R}^n$-valued ones'.
Since, the Borel sets can be formed from the sets of the form $( -\infty , x)$ by using the operations of taking complements, countable union and countable intersections, so the above property in terms of CDFs is equivalent to the definition of independence.