Concentration Properties of Restricted Measures with Applications to Non-Lipschitz Functions

We show that for any metric probability space $(M,d,\mu)$ with a subgaussian constant $\sigma^2(\mu)$ and any set $A \subset M$ we have $\sigma^2(\mu_A) \leq c \log\left(e/\mu(A)\right)\,\sigma^2(\mu)$, where $\mu_A$ is a restriction of $\mu$ to the set $A$ and $c$ is a universal constant. As a consequence we deduce concentration inequalities for non-Lipschitz functions.