Aubin--Lions Compactness for Stieltjes--Bochner Evolution Graphs

Let $g$ be a nondecreasing left-continuous function and let ${\mu }_g$ be its Lebesgue–Stieltjes measure. We establish a Banach-valued fundamental theorem of calculus and an Aubin–Lions compactness theorem for evolution measured by ${\mu }_g$, allowing absolutely continuous, singular continuous and atomic components. If the range space has the Radon–Nikodým property, a curve is $g$-absolutely continuous if and only if it is an indefinite Bochner integral; its strong $g$-derivative is the Bochner density and the variation measure has density equal to its norm. A terminal atom may make the derivative invisible from the Bochner state class, so the natural evolution space is a graph of state–derivative pairs. For $B_0⋐B\hookrightarrow B_1$, with $B_0$ and $B_1$ reflexive and $1<p_0,p_1<\mathrm{\infty }$, the state projection is a compact linear operator into $L_g^{p_0}([a,b);B)$. The proof combines bounded evaluation at atoms, local ${\mu }_g$-averages at nonatomic points and Ehrling’s inequality. The result recovers the classical theorem and yields compactness principles for bounded time scales, weighted sequences and mixed Stieltjes measures.