Universal metric proper G-spaces

It is proved that for G a locally compact group of weight less than or equal to a given infinite cardinal number τ, there exists a metric proper G  -space J∞(G,τ)J∞(G,τ) of weight τ which is a G-AR and every metrizable proper G-space X of weight ≤τ   can equivariantly be embedded in J∞(G,τ)J∞(G,τ). As a by-product we prove that every compact subgroup of a locally compact group G is contained in a maximal compact subgroup of an open almost connected subgroup of G. This property is responsible for equivariant extension properties of the universal proper G  -space J∞(G,τ)J∞(G,τ).