  This article studies a priori error analysis for linear parabolic interface
problems with measure data in time in a bounded convex polygonal domain in
$\mathbb{R}^2$. We have used the standard continuous fitted finite element
discretization for the space. Due to the low regularity of the data of the
problem, the solution possesses very low regularity in the entire domain. A
priori error bound in the $L^2(L^2(\Omega))$-norm for the spatially discrete
finite element approximations are derived under minimal regularity with the
help of the $L^2$ projection operators and the duality argument. The interfaces
are assumed to be smooth for our purpose.
