Is this a probability on $[0,1]$? Given the domain $\Omega = [0,1]$ and the correpsonding Borelian $\sigma$-algebra $A = B([0,1])$ I have to find a probability measure $P$ such taht $(\Omega, A,P)$ is a probability space. I think I have found such an example but I'm not completely sure:

Let $P(\{x\}) := \begin{cases} 1 & x=1 \\ 0 & x\neq 1\end{cases}$ or equivalently $P(X) := \begin{cases} 1 & 1 \in X \\ 0 & x\not\in X\end{cases}$ for $X \in A$.

I checked the axioms:


*

*$P$ is a well defined function $P:A \to [0,1]$.

*$P(\Omega) = 1$ since $1 \in [0,1]=\Omega$.

*$P$ is obviously $\sigma$-additive


But still I have the feeling that something is wrong. So can you tell me whether this is correct or I did something wrong?

Your solution is correct with one remark: the first definition does not determine $P$ even if we try to naturally extend the definition using countable additivity:
$$P(B) = \bigcup_{b \in B} P( \{ b \} )$$
since it only works for countable $B$, leaving $P(B)$ undefined for uncountable $B \in A$.
The second definition is correct and should be the one used.
