C-embeddings of spaces of signed measures

For a Tychonoff space X let Mₐ(X)⊃ M_σ(X)⊃ M_τ(X)⊃ MR(X) be the spaces of finitely additive, $σ$-additive (Baire), $τ$-additive and Radon signed measures on X, and let M_β(X) and Md(X) be the spaces of compactly supported and discrete measures; all of them are regarded as subspaces of M(βX)=C(βX)^* with the weak^* topology. We study when Mₚ(X) is C-embedded (C^*-embedded) in Mq(X). We prove that C-embeddedness, C^*-embeddedness and G_δ-density of Mₚ(X) in Mq(X) are equivalent, and that they are equivalent to a lifting property: for every continuous map f of X into a metrizable compact space, the image of every probability measure in Mq(X) is the image of a probability measure in Mₚ(X). The main tool is a general theorem, proved in a separate paper, asserting that C-embeddedness, C^*-embeddedness and G_δ-density are equivalent for dense subspaces of locally convex spaces in the weak topology, and, more generally, of products of first-countable spaces. We characterize the spaces X for which M_σ(X), M_β(X), Md(X) and MR(X) are C-embedded in Mₐ(X) (pseudocompactness, pseudo-$ω$-boundedness, scatteredness of $βX$, and a lifting property of Radon measures, respectively). We give an example of a pseudocompact locally compact space X for which M_τ(X) is not C^*-embedded in M_σ(X), and we show that M_τ(X) is C-embedded in $M(βX)$ for every G_δ-dense subspace X of a product of metrizable compact spaces. Using a set of universal measure zero, we construct a pseudocompact space X for which MR(X) is C-embedded in $M(βX)$, although the restriction to X of a coordinate projection onto $[0,1]$ maps no $σ$-compact subset of X onto $[0,1]$.