# What do the notations $\sim$ and $\Delta (A)$ mean in the paper "Fairness Through Awareness"?

In this paper Fairness Through Awareness, the notation $$\mathbb{E}_{x \sim V} \mathbb{E}_{a \sim \mu_x} L(x,a)$$ is being used (page 5 top line), where $$V$$ denotes the set of individuals (so I guess set of feature vectors?) and the meaning of the other variables can be found in the paragraph above the mentioned notation. What does the $$\sim$$ in the expectation stand for?

Another notation that I do not know is $$\Delta (A)$$, where $$A$$ is the set of outcomes, for instance, $$A = \{ 0,1\}$$. What does it stand for?

The $$\sim$$ symbol means that a random variable is drawn from the given distribution, i.e. if I were to say $$X$$ has a Standard Normal distribution I would write $$X \sim \text{Normal}(0,1)$$. They write two explicit expectations here because $$a$$ is a random variable with distribution $$\mu_x$$ but $$X$$ is also a random variable with distribution $$V$$. I believe you are right that $$V$$ would be analogous to a set of features. So we are saying that $$X$$ is a random variable over the distribution over the features, or individuals in this context.

As for the $$\Delta(A)$$, I have never seen this notation used before -- I am not sure if it is standard notation. I Googled to see if it was something that I just hadn't seen before, but there was no such answer. However, from the context of the paper they define $$M: V \rightarrow \Delta (A)$$ to be mappings from individuals to probability distributions over $$A$$, so I guess that $$\Delta (A)$$ is probability distributions over $$A$$.