Just in the interest of understanding the "invariant measure" on Minkowski space used in QFT calculations. The point is that the usual measure function is chart dependent, in other words different coordinate systems disagree on the measure value of measurable sets, but the "invariant measure" means different coordinate systems agree on the output of the measure function these sets?

what would "the usual measure" be?

$d^4x$?

that's a measure on a different manifold though, isn't it?

and it's also Lorentz invariant btw

wait what manifold is that measure on?

oh maybe I'm thinking of $d^4p$

sorry, to make sure we are talking about the same thing: I think of the "Lorentz invariant measure" as that used to integrate over a 3D sub-space of fixed norm ("on shell")

ok mb I looked up what I was talking about, I mean: $$\int d^3p\rightarrow \int \frac{d^3p}{2E_p}.$$

as in $d^3 k / E(k)$ (up to some numbers)

yes, it's a 3D integral, you're not integrating over the whole 4D space

you can get it from the 4D measure as $\delta(p^2-m^2)d^4p$

I recall having seen the derivation at some point

it's just a change of variables in the delta

anyway yes, it's invariant in the sense that it's the same in every frame, as opposed to $d^3 k$ which would change

ok thanks