The Nikodym property in the Sacks model

We prove that if A is a σ-complete Boolean algebra in a ground model V of set theory, then A has the Nikodym property in every side-by-side Sacks forcing extension V[G], i.e. every pointwise bounded sequence of measures on A in V[G] is uniformly bounded. This gives a consistent example of a class of infinite Boolean algebras with the Nikodym property and of cardinality strictly less than the continuum.