Suppose that $X$ is a locally compact Hausdorff space. For a Radon measure $\mu$ on $X$, let $I_{\mu}\colon C_{c}(X)\to\mathbb{C}$ be the positive linear functional defined by $I_{\mu}(f):=\int_{X}f \ \text{d}\mu$. The Riesz representation theorem imples that the assignment $\mu\mapsto I_{\mu}$ implements a one-to-one correspondence between (positive) Radon measures on $X$ and positive linear functionals on $C_{c}(X)$. So one could define an integral on $X$ as a positive linear functional $I\colon C_{c}(X)\to\mathbb{C}$. This definition does not rely on measure theory. I was wondering whether we could prove Fubini's theorem in this setting, i.e. without refering to measure theory.
More precisely, does anyone know a proof or reference of the following statement without measure theory?
Let $I$ and $J$ be positive linear functionals (i.e. integrals) on locally compact Hausdorff spaces $X$ and $Y$, respectively. For $f\in C_{c}(X\times Y)$ and $y\in Y$ we define $f^{y}\colon X\to\mathbb{C}$ via $f^{y}(x):=f(x,y)$. For $x\in X$ we define $f_{x}\colon Y\to\mathbb{C}$ similarly. The functions $x\mapsto J(f_{x})$ and $y\mapsto I(f^{y})$ are compactly supported and $$I(x\mapsto J(f_{x}))=J(y\mapsto I(f^{y})).$$
Here is a sketch for the proof: The result is clear for separable functions $f$ of the form $f(x,y) = g(x) h(y)$. The span of such functions should be dense in $C_c(X \times Y)$. Thus, we can approximate all functions in $C_c(X \times Y)$ by sums of separable functions and this gives the result.