Is $\mathbb{P}((a,b)) = \tan^{-1}(b) - \tan^{-1}(a)$ a valid probability measure on $X = (0, \infty)$ ?
I recently came across this question and thought that this is not a valid probability measure as the this will map the entire sample space interval to $\frac{\pi}{2}$.
It is given that $\mathbb{P}$ is a function that maps $\sigma$ algebra to the $\mathbb{R}^+$. So I thought that we can form a Borel algebra on $X = (0, \infty)$, then try to prove the given function is indeed a probability measure or not.
While the above function satisfies the $1^{st}$ axiom of probability measure, for the 2nd axiom: $\mathbb{P}((0,\infty)) = \tan^{-1}(\infty) - \tan^{-1}(0) = \frac{\pi}{2} \neq 1 $. Hence, I thought $\mathbb{P}$ is not a valid probability measure. Also, the range of $\mathbb{P}$ is outside $[0,1]$. But, it was said to be a valid probability measure without giving any reason in the solution.
Is this function a valid probability measure, if so how does this satisfy the $2^{nd}$ axiom?