I'd like to find a probability distribution $f(x)$ on the unit interval $[0,1]$ that obeys a given set of moment constraints, e.g. $\int_0^1 xf(x) dx = \mu_0$ for some given $\mu_0$, and so forth. I'd like the distribution for which $\int_0^1 \sqrt{f_c(x)} dx$, where $f_c$ is the absolutely continuous part of $f$, is as large as possible. What vector space should I consider this over? Should it be $L_1$?
As an example, if I only know a constraint on the first moment $\int_0^1 xf(x) dx = \mu_0$, then by discretizing $f$, I would "guess" that the worst-case distribution takes the form $f(x) = \frac{1}{4(\lambda_1+\lambda_2x)^2}$ for suitable $\lambda_i$'s. Is there a way to make this argument formal?
If you have all the $\left(\mu_n\right)_{n=0}^\infty$, you can constrain the moment generating function and optimize your $\int \sqrt{f_c(x)} dx$. You should make sure the integral exists, that would determine the underlying vector space...