 Hypergraph degree measure

For a uniform hypergraph with positive average hypergraph vertex degree $d$, the degree measure is $\mu (S)=(nd{)}^{-1}{\sum }_{u\in S}{d}_H(u)$. It is a probability measure and equals $|S|/n$ in a regular hypergraph. Every hypergraph independent set has degree measure at most $1-1/r$, even when its fraction of all vertices is close to one.