Integrating the composition of a Heaviside function with a smooth function

Question

I am trying to find how to compute an integral of the form: $\int_{R^n}{\Theta(g(x))f(x)\,dx}$, where $\Theta$ is the Heaviside function, $g(x)$ is a smooth function (a result for more general $g$ is also welcome) and you can assume that $f(x)$ is also smooth. For the Dirac delta (which is the derivative of $\Theta$) we have the relationship: $$\int_{R^n}{f(x)\delta(g(x))|\nabla g(x)|\,dx} = \int_{R^n}{f(x)\delta_S(x)\,dx} = \int_{S}{f(x)\,d\sigma(x)}$$ Where $S = \{x|g(x) = 0, x \in R^n\}$, and $\sigma(x)$ is the surface measure on $S$. Is there a similar relationship for the Heaviside function? Or any standard method that would help me compute integrals of that form. References on the subject are welcome, but keep in mind I am a computer science student (so my mathematical background is fairly limited).

Answer 1

Turns out there is no such relationship for the Heaviside step function (and it's not necessary). We can directly apply it to the integration domain. That is: $$\int_{R^n}{\Theta(g(x))f(x)\,dx} = \int_{\{x|g(x)\geq 0\}}{f(x)\,dx}$$