We introduce the category \(\FVA\) of finite-valuation approximable domains, a full subcategory of continuous domains contained in the category of pointed countably based FS-domains. We prove that \(\FVA\) is Cartesian closed and closed under both the subprobability and probability valuation powerdomains. Hence the valuation monads \(\Vsub\) and \(\Vone\) restrict to \(\FVA\), yielding a positive answer to the generalized form of Jung--Tix problem, one of the longest-standing open problem in domain theory since 1990s. The proof is divided into two steps. First, for every finite poset \(P\), we construct an increasing FS approximate identity on \(\Vsub(P)\), and thereby show that \(\Vsub(P)\) is a countably based FS-domain. Second, we call a domain finite-valuation approximable when its identity is the pointwise supremum of an increasing sequence of maps factoring through spaces \(\Vsub(P_n)\), where each \(P_n\) is finite. A finite-separation saturation theorem and a unified kernel-lifting theorem then show that \(\FVA\) is closed under Scott-continuous retracts, finite products, function spaces, \(\Vsub\), and \(\Vone\).

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