My question conerns Example 1 in §9.3 of Luenberger's Optimization by Vector Space Methods. There, he considers the problem of finding $x\in D^n(t_o,t_1)$ (n-vector functions possessing continuous derivatives with fixed end points $x(t_0), x(t_1)$) minimizing
$\int_{t_0}^{t_1} f(x,x',t) \,dt $
with the constraint
$\phi(x,t) = 0$,
where $\phi$ is a functional mapping from a subspace of $D^n(t_0,t_1)$ into the subspace $Y$ of $D(t_0,t_1)$ with functions vanishing at $t_0$ and $t_1$.
The multiplier of the constraint (element of $Y^\ast$) can be represented by
$\int_{t_0}^{t_1} y'(t)\, dz(t) $
with $z$ being a function of bounded variation on $[t_0, t_1]$. The author then states that in this setting the multiplier can be rewritten as
$\int_{t_0}^{t_1} y(t)\,\lambda(t)\, dt $
with a continuous function $\lambda$.
I am trying to proof this claim and I am stuck. I started with basic properties of Stieltjes integrals, e.g. assume that $z$ is differentiable with integrable derivative. But I do not see why this should be satisfied in this setting. Integration by parts doesn't help either as far as I can see. Can someone proof this claim or point me in the right direction? Are there more general statements concerning the expression of a Stieltjes integral (in the setting of Lagrange multipliers) as a Lebesgue integral. It seems to me that in the area of calculus of variations this result is commonly used without any proof. Unfortunately, I am new to this topic.