Pigeonhole principle in terms of measure of sets

Question

I'm wondering if this pigeonhole-like statement holds. Let $(S_i)_{i \in N}$ be a countable family of subsets of $[0,1]$ such that each $S_i$ is (either Jordan or Lebesgue) measurable and there is some $\delta$ s.t. each $S_i$ has measure at least $\delta$. Is it true that there is some infinite subset $I \subset N$ s.t $\cap_{i \in I}S_i$ is nontrivial? Would this depend on the measure used/in case this doesn't hold, could additional assumptions on the sets be added to make it true?