In Srednicki's QFT book, in chapter 3 (eqn. 3.16 onwards) he talks about the lorentz invariance of the volume element. For this he writes $d^3k/f(k)$ should be invariant under lorentz transformations. He then tells us that an integration measure that's manifestly invariant under such a transformation is $d^4k \delta (k^2+m^2)\theta(k^0)$.
I know that $d^4k$ is manifestly invariance under lorentz transformations since $det \Lambda = 1 or -1$. Why do we multiply the on-shell condition and the theta function to it? And then why do we integrate over $k^0$ to finally obtain $f(k)\propto \omega$?
Why can't we simply write $d^3k = d^4k/E_k$ transforming from the spatial element to a space-time element?
I'm looking for rationale (physical and mathematical) for going through the particular hoops that Srednicki does.