  We study the nonlinear stochastic heat equation driven by space-time white
noise in the case that the initial datum $u_0$ is a (possibly signed) measure.
In this case, one cannot obtain a mild random-field solution in the usual
sense. We prove instead that it is possible to establish the existence and
uniqueness of a weak solution with values in a suitable function space. Our
approach is based on a construction of a generalized definition of a stochastic
convolution via Young-type inequalities.
