Vtyjkyuk

What can be a possible probability measure on $(0,infty)$? Give an example.

For $(0,1)$ Lebesgue measure can be used and it easily satisfies all the properties of probability measure. But when the set is $(0,infty)$, Lebesgue measure will not lie in 0 to 1 range. I am thinking that some mapping from $(0,infty)$ to $(0,1)$ would do the trick. Am I right?

Any Lebesgue integrable function $f:(0,infty)to[0,infty)$ which does not vanish almost everywhere can be made into a probability measure $mu_f$ on $(0,infty)$ by setting $$mu_f(S):=fracint_S,f(x),textdxint_0^infty,f(x),textdxtext for every measurable set S,.$$

Every absolutely continuous probability measure (relative to the Lebesgue measure) arises this way.

However, there are uncountably many other probability measures. Singular continuous measures on $(0,infty)$ such as the Cantor distribution and discrete probability measures on $(0,infty)$ are some of those probability measures not in the form $mu_f$ for some Lebesgue integrable function $f$. Of course, you can also have a convex combination of an absolutely continuous probability measure, a singular one, and a discrete one.

Additionally, if you want a probability measure $nu$ whose essential support is precisely $(0,infty)$, then the absolutely continuous part $nu_textabs$ of $nu$ cannot be $0$ (since essential supports of singular and discrete probability measures are Lebesgue null sets). In other words, $nu_textabs$ is of the form $nu_textabs=alpha,mu_f$ for some $alphain(0,1]$ and for some Lebesgue integrable function $f:(0,infty)to[0,infty)$ which vanishes on a set of Lebesgue measure $0$.