Coherency w VaR and CTE

I had a question on testing coherency for potential risk measures if that came up on the exam...



Properties of coherency:

Monotonicity: If X <= Y then f(X) <= f(Y)

Subadditivity: f(X+Y) <= f(X) + f(Y)

Positive homogeneity: lambda*f(X) = f(lambda*X) for lambda > 0

Translation invariance: a + f(X) = f(X + a)



How exactly do we interpret X<=Y and X+Y if we are talking about a set of numbers, since VaR and CTE are normally calculated on a set of numbers?



For example if I have the set X = {1,2,3} and the set Y = {4,5,6}, what does X<=Y and X+Y really mean in this case? While we're at it, would lambda * X and a + X simply be multiplying the set of numbers by lambda and adding a to all numbers in the set?



Thanks!

Coherency w VaR and CTE