  We study the solutions of the stochastic heat equation driven by spatially
inhomogeneous multiplicative white noise based on a fractal measure. We prove
pathwise uniqueness for solutions of this equation when the noise coefficient
is H\"{o}lder continuous of index $\gamma>1-\frac{\eta}{2(\eta+1)}$. Here
$\eta\in(0,1)$ is a constant that defines the spatial regularity of the noise.
