In case of an almost surely continuous function $f$ we know that the Lebesgue Measure coincides with the Riemann Integral.
With this in mind, supposing $\mathbf{X}$ is a rough path lift of $X$, is there some measure $\Lambda_\mathbf{X}$ such that $$\int_0^t f(X_s) d\mathbf{X}_s = \int_0^t f(X_s) d\Lambda_\mathbf{X}(s)$$ where the left hand side integral is the rough path integral of a finite $p$-variation path with $p \in (2,3]$?
If so, what is a possibile construction of $\Lambda_X$?
To be clear there is no universal measure $\Lambda$ that can represent any given rough path.
A better analogy is with Riemann-Stieltjes integrals or even better Young integrals $\int f dg$ with $f\in C^{p},g\in C^{q}$ with $p+q>1$ (since after all Rough paths are a generalization of them).
Here the measure is the total-variation measure $dg$
However, generally for rough paths there is not measure necessarily. For example for Itô integrals (which are part of the rough-path framework), we have that $dB_{t}(\omega)$ is not a regular measure see What kind of "measure" is $dB_{t}(\omega)$ because the Itô integrals are interpreted as $L^{2}$-limits (since B has infinite first variation).