For the heat equations driven by a stochastic measure on $[-𝐿,𝐿]$ with the Dirichlet boundary condition, we prove that solutions tend to the solution of the heat equations defined on R as $𝐿\to \mathrm{\infty }$. The estimate of the convergence rate is obtained. For a stochastic measure, we assume the 𝜎-additivity in probability only.