Let $X$ be a compact metric space, $\mu$ a finite Borel measure on $X$. Let $C(X)$ be the Banach space of complex-valued continuous functions on $X$, equipped with the maximum norm. Suppose that $k \in C(X \times X)$ and consider the integral operator $K : C(X) \to C(X)$ defined by
$$C(X) \ni f \mapsto \int_Xk(\cdot, y)f(y)d\mu(y).$$
One can use the Arzela-Ascoli theorem to show that $K$ is a compact operator $C(X) \to C(X)$. Thus we can apply the Riesz-Fredholm-Schauder theory to the operator $I - \lambda K$, $\lambda \in \mathbb{C}\setminus \{0\}$. That is, we know that
$$\dim N_{I - \lambda K} = \dim N_{I - \lambda K^*}< \infty, $$ $$ R_{I - \lambda K} = N^\perp_{I - \lambda K^*}, \qquad R_{I - \lambda K^*} = N^\perp_{I - \lambda K},$$
where $N$ denotes nullspace, $R$ denotes range, $K^*$ denotes adjoint, and $\perp$ denotes orthogonal complement.
Let $\varphi_i$, $i =1, \dots, m$ be a basis for the nullspace of $I - \lambda \tilde{K}$, where $\tilde{K}$ is defined by the kernel $\tilde{k}(x,y) = k(y,x)$. If $g \in C(X)$ has the property that $\int_X g(y)\varphi_i(y) d\mu(x) = 0$, $i =1 ,\dots, m$, prove that $g$ is in the range of the operator $(I - \lambda K)$.
This is part of problem 5.2.3 from Friedman's Foundations of Modern Analysis.
My attempt so far: because we have $\int g\varphi_i = 0$, and the $\varphi_i$ are a basis for $N_{I - \lambda \tilde{K}}$, we know that the functional $G \in (C(X))^*$ defined by $$ C(X) \ni h \mapsto G(h) := \int_Xg(y)h(y) d\mu(y)$$
belongs to $N^\perp_{I - \lambda \tilde{K}}$. So by the Riesz-Fredholm-Schauder theory, we have $G \in R_{I - \lambda \tilde{K}^*}$. Therefore, there exists an element $F \in (C(X))^*$ such that
$$F((I - \lambda \tilde{K})h) = \int_Xg(y)h(y)d\mu(y), \qquad \forall h \in C(X).$$
I've gotten stuck at this point. I need to somehow use this functional $F$ to produce a function $f \in C(X)$ so that $(I - \lambda K)f = g$. I'm wondering if the problem may need some additional assumptions, such as $\mu$ needing to be a Radon measure so that some sort of representation theorem or Radon-Nikodym-type theorem can be applied. Hints or solutions, and perspective are greatly appreciated!