Let's consider the Lebesgue measure ($\mu$) over the closed interval $[0,1]$. As you know, $\mu(\mathbb{Q} \cap [0,1]) = 0$.
In other way, as far as I know the computer just can represent accurately a countable number of irrational numbers (see this blog).
So my question is, how can we prove assertions about measures (I am thinking in dynamical systems) if we cannot generate initial conditions with measure different of zero.
Note: Relax "prove assertions about measures" in the last paragraph by "determine qualitatively invariant measures".
Proving things mathematically does not depend nor need a computer.
But you are right in the sense that simulating dynamical behaviour in a computer often is difficult. The amount of accuracy of a real world computer is limited, so one always has to approximate a dynamical system by a finite state space and finite computational capacity. For example if you have a dynamical system which is sensitive to initial conditions, approximating the starting value (which is necessary given the finite accuracy of a computer) might drastically change the long-term behaviour. Thus simulations always have to be accompanied by error estimates and careful analytic thinking.