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Fix a base ring $R$, and consider pairs $(X,f)$ where $X$ is a scheme of finite type over $R$ and $f:X\to R$ is an $R$-valued algebraic (not constructible!) function on $X$.

I want to consider a Grothendieck $R$-algebra of such pairs, where if $X = Y \coprod Z$, then $[(X,f)] = [(Y,f|_Y)] + [(Z,f|_Z)]$, but also $[(X,f+g)] = [(X,f)] + [(X,g)]$ and $[(X,rf)] = r[(X,f)]$.

Surely this is a standard extension of the usual notion of the Grothendieck ring of varieties (which only has $f=1$, and the first sort of relation)? If so, where can I read about it?

Maybe I'm misreading the motivic integration survey literature (by K. Smith, and E. Looijenga), but it seems like they're insisting on constructible functions, not algebraic. Ordinarily when a construction like this isn't in the literature, I assume it's because it has too many relations and is $0$, but if $R = {\mathbb Z}$ it seems to me that this ring has many functionals, like $[(X,f)] \mapsto \sum_{x \in X_p} (f(x) \bmod p) \in {\mathbb Z}/p.$ (I don't see an analogue of $[X] \mapsto$ the Euler characteristic $\chi(X_{\mathbb C})$.)

EDIT: One problem I see is that $({\mathbb A}^1, f(x)=x)$ is isomorphic under translation to $({\mathbb A}^1, f(x)=x+1)$. So $[({\mathbb A^1}, 1)] = [({\mathbb A}^1, (x+1)-x)] = [({\mathbb A}^1, x+1)] - [({\mathbb A}^1, x)] = 0$. Of course this fits with point-counting $\bmod p$.

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# Grothendieck ring of "varieties carrying a function"

Fix a base ring $R$, and consider pairs $(X,f)$ where $X$ is a scheme of finite type over $R$ and $f:X\to R$ is an $R$-valued algebraic (not constructible!) function on $X$.

I want to consider a Grothendieck $R$-algebra of such pairs, where if $X = Y \coprod Z$, then $[(X,f)] = [(Y,f|_Y)] + [(Z,f|_Z)]$, but also $[(X,f+g)] = [(X,f)] + [(X,g)]$ and $[(X,rf)] = r[(X,f)]$.

Surely this is a standard extension of the usual notion of the Grothendieck ring of varieties (which only has $f=1$, and the first sort of relation)? If so, where can I read about it?

Maybe I'm misreading the motivic integration survey literature (by K. Smith, and E. Looijenga), but it seems like they're insisting on constructible functions, not algebraic. Ordinarily when a construction like this isn't in the literature, I assume it's because it has too many relations and is $0$, but if $R = {\mathbb Z}$ it seems to me that this ring has many functionals, like $[(X,f)] \mapsto \sum_{x \in X_p} (f(x) \bmod p) \in {\mathbb Z}/p.$ (I don't see an analogue of $[X] \mapsto$ the Euler characteristic $\chi(X_{\mathbb C})$.)