Bernstein–Markov Property for Compact Sets in $$\mathbb {C}^d$$

Given a compact set K in \(\mathbb {C}^d.\) We concern with the Bernstein–Markov property of the pair \((K,\mu )\) where \(\mu \) is a finite positive Borel measure with compact support K. In particular, we are able to give a class of \((K,\mu )\) having the Bernstein–Markov property with the measure \(\mu \) satisfies a rather weak density condition. Using this result, we construct a pair \((K,\mu )\) satisfying the Bernstein–Markov property which is not covered by the known results in Bloom (Indiana Univ Math J 46:427–452, 1997) and Bloom and Levenberg (Trans Am Math Soc 351:4573–4567, 1999). Another main result of the note is a weak characterization of Bernstein–Markov property in terms of Chebyshev constants.