Modeling the idea of "equally-likely" in a continuous world requires a slightly different perspective since there are obviously infinitely many outcomes to consider. Instead, you should consider requiring that intervals in the domain which are of equal width should have the same probability regardless of where they are in that domain. This behaviour suggests
\begin{equation*}
P(u \lt X \lt v) = P(u + \Delta \lt X \lt v + \Delta)
\end{equation*}
for reasonable values of \(\Delta\) so that the interval remains inside \(R\text{.}\)