Is probability that $x<2$ equal to the probability that $x^2<4$ given $1Assuming $x$ is a real number uniformly distributed over the interval $(1,3).$
so $x^2$ is also uniformly distributed over the interval $(1,9)${As for every $x=a\in (1,3) $ there exists $x^2=a^2\in (1,9)$}.
Probability that $x<2$ would be $\frac{1}{2}$ as $x$ can be in $(1,2)$ where sample set of $x$ is $(1,3)$, while probability that $x^2<4$ is $\frac{3}{8}$ as $x^2$ can be in $(1,4)$ where sample set of $x^2$ is $(1,9).$
So why is the probability that $x<2$ different from $x^2<4$ if both are identical?

You decided for any $x$ to match $x^2$, so you are claiming that $x<2$ is identical to $x^2<4$, but you can also match $4x-3$ in the interval $(1,9)$ for all $x$ in the interval $(1,3)$, which makes $x<2$ the same as $4x-3<5$.
That is the problem of defining probability on infinite group.
