In a continuous optimization problem, I consider the functions belonging to the "space of probabilities" on $\lbrack 0,1 \rbrack^N $ that admit a probability density. I understand that I exclude in the process distributions such as the Dirac.
Can I use a subset of the Lebesgue $L_1(\lbrack 0,1 \rbrack^N) $ space for that matter? To be specific, can I use : $$ \{ p \in L_1 | \int p = 1, p\geq0 \} $$