Proposition 2

The support of the product $\lambda \otimes \mu$ is equal to the product of the support of $\lambda$ and the support of $\mu$.

We first observe that the relation $\lambda \otimes \mu = 0$ implies that one of the measures $\lambda, \mu$ is zero (A, II, §7, No. 7, Prop. 16, (ii)). On the other hand, if U (resp. V) is an open set in X (resp. Y), then the restriction of $\lambda \otimes \mu$ to the product $U \times V$ is the product of the restrictions of $\lambda$ to U and of $\mu$ to V, as follows from Th. 1 of No. 1 and the definition of the restriction of a measure to an open set (§ 2, No. 1). It therefore follows that, for the restriction of $\lambda \otimes \mu$ to $U \times V$ to be zero, it is necessary and sufficient that either the restriction of $\lambda$ to U or the restriction of $\mu$ to V be zero, which proves the proposition, on taking into account the definition of the support of a measure (§ 2, No. 2).