I am trying to solve the following equation $$\int_{-\infty}^{+\infty} \Theta(g(\tau))f(\tau)d\tau$$ with $\Theta(t)$ being the heaviside step function defined by: $$\left\{\begin{matrix} \Theta (t) =0, t<0 \\ \Theta (t) =1, t\geq 0 \end{matrix}\right. $$ and $g(t)$ a continuously differentiable function and $f(t)$ a test function.
Knowing the link between the Heaviside and the Dirac functions, I was wondering if we could get to the same kind of property as for the Dirac in this case: $$ \int_{-\infty}^{+\infty}\delta (g(\tau))f(\tau)d\tau = \sum_{i} \frac{f(x_{i})}{\left | g'(x_{i}) \right |}. $$
Or if you have any other idea of how to solve it, it would be great!