Corollary 2

Let $u_1$ and $u_2$ be two locally $\theta$-integrable complex functions. In order that $u_1 \cdot \theta = u_2 \cdot \theta$, it is necessary and sufficient that $u_1$ and $u_2$ be equal locally almost everywhere.

One is immediately reduced to showing that $u \cdot \theta = 0$ implies that $u(t) = 0$ locally almost everywhere; but $u \cdot \theta = 0$ means that the function 1 is locally negligible for the measure $u \cdot \theta$. One then applies Corollary 1.