Constructing a weird probability measure

Is it possible to construct a probability space $$(\Omega,\mathcal{A},\mathbb{P})$$ such that $$\mathcal{A}$$ is uncountable, there are uncountable events with probability $$>0$$ and there are also countably many measurable singleton $${x}$$ that have probability $$>0$$?

This seems like a really weird probability measure to have - and I wasn't able to construct one. All non-toy example with uncountable $$\mathcal{A}$$ that I know assign measure $$0$$ to singletons.

• Hint: take a probability measure $\mu$ with the first property, a probability measure $\nu$ with the second one, and consider $1/2(\mu+\nu)$ – Lucio Feb 16 at 20:19

Why be fancy? A countable set can have uncountably many subsets. For example, take $$\Omega=\mathbb N$$ and $$\mathcal A=2^\Omega$$ and $$P(X=n)=2^{-n}$$.
Let $$\mu=\frac 1 2 \nu+\frac 1 2\sum \frac 1 {2^{n}} \delta_n$$ where, for example, $$\nu$$ is the $$N(0,1)$$ measure. [$$\mu(x,x+1)>0$$ for all $$x$$ and $$\mu\{n\}>0$$ for all $$n$$].
Take a right continuous function $$F: \mathbb{R} \to \mathbb{R}$$ with $$F(1)- F(0) = 1$$ that has countably many discontinuities in $$]0,1]$$ and consider the measure space $$(]0,1], \mathcal{B}(]0,1]), \mu_F)$$ where $$\mu_F$$ is the Lebesgue-Stieltjes measure restricted to the Borel sets on $$]0,1]$$.