Probability of random vector lying on a hyperplane

Question

I have a random vector $v \in \mathbb R^n$, of which the elements are independent. Now there is also a hyperplane $S \subseteq \mathbb R^n$ of dimension $n-1$. The vector is drawn from any continuous probability distribution. Now my common sense tells me that the probability that the vector lies on the hyperplane, is zero ($P(v\in S)=0$). But how would I prove this? And is this even true?

Answer 1

Yes, the probability is zero. You prove it by introducing an integration measure and calculating the integral.