Is $L^2(\mathbb R, \mathcal P(\mathbb R), \#)$ well defined?

Question

Is $L^2(\mathbb R, \mathcal P(\mathbb R), \#)$ well defined ? $\#$ is the measure that counts the points. If so, is $L^2(\mathbb R, \mathcal A, \#)$ the same as $L^2(\mathbb R, \mathcal A, \lambda)$ where $\lambda$ is the Lebesgue, and $\mathcal A$ the Lebesgue $\sigma$-algebra ?

Answer 1

Of course, $L^{2}$ is defined for any measure. In this case $L^{2}$ consists precisely of all functions $f$ such that there is an at most countable set $(x_n)$ with $f(x)=0$ for $x \notin (x_n)$ and $\sum |f(x_n)|^{2} <\infty$. The answer to the second question is obviously no.