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

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

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