Confusing notation in the definition of differential privacy
I had the same problem when I was understanding this because it's not so intuitive. I will answer here in case future people had the same trouble.

Agree with you $\Delta B$ assigns a probability to every element of the set $B$.

$A$ It can be anything I assume from continuos to discrete objects.

$(M(a))_b$ is the probability that the mechanism outputs $b$ when the input is $a$.

The probability space is over the coin flips of the algorithm $\mathcal{M}$. This means you will not know before you apply the mechanism which will be the result. For example, if you toss a coin, you will know what will be the answer? No. What this space is inducing the most important, the randomness. You will see on some papers &quot; The probability space is over the randomness of the algorithm $\mathcal{M}$&quot;, which makes more sense.


I made an example to stabilize these definitions:

Let's consider an mechanism $\mathcal{M}$ that will find a $a$ in a list $A=\{(x_0,x_1,x_2,x_3,...,x_n): x_i=\{a,b\},n\geq 2\}$ where the half are a's and the other half are b's. With output $B=1$ if it finds an a and $B=0$ if it doesnt . This is $B=\{0,1\}$. Then the algorithm/mechanism $\mathcal{M}(x_i)=b$ with probability $(M(x_i))_b)$ which is defined by $\{(p_1,p_2)\in \mathbf{R}^2: p_1+p_2 = 1;p_1,p_2 \geq 0\}$ (Probability Simplex) . It means give a value of $p_1$ that $B=1$ happens and $p_2$ that $B=0$ happens . Who define this probabilities ?. The space of randomness, which we will never know for sure will be the result of the mecanism, because in that case wil be deterministic.