In this thesis , we present an expanded account of the work " Probability measures on [SIN] groups and some related ideals in group algebras " by Jaworski . Let G be a locally compact second countable group . Given a probability measure \\mu on G , the set J_{\\mu} = \\overline{{ \\varphi - \\varphi * \\mu \\varphi \\in L^{1}(G) } }, where the bar means closure in the L^{1} norm , is a left ideal in the group algebra L^{1}(G) . The annihilator of J_{\\mu} in L^{\\infty}(G) is a weakly* closed suace H_{\\mu} of L^{\\infty}(G) , containing constants and invariant under the action of G by left translations . The elements of H_{\\mu} are called bounded \\mu -harmonic functions . They are precisely those elements h \\in L^{\\infty}(G) which satisfy h(g) = \\int_{G}h(gg')\\mu(dg') (modulo the Haar measure) . The questions which we consider in this work have to do with the dependence of J_{\\mu} (and H_{\\mu} ) on \\mu . While J_{\\mu} and H_{\\mu} are completely determined by \\mu , it is well known that the correspondence \\mu \\rightarrow J_{\\mu} (and \\mu \\rightarrow H_{\\mu} ) is far from one to one . As a matter of fact , for many groups , including abelian groups and compact groups , J_{\\mu} and H_{\\mu} depend only on the closed subgroup , G_{\\mu} , generated by the support of \\mu : J_{\\mu} = L_{0}^{1}( G , G_{\\mu} ) = the kernel of the canonical mapping from L^{1}( G ) to L^{1}( G/G_{\\mu}) , and H_{\\mu} = L_{0}^{\\infty}( G , G_{\\mu} ) = { f \\in L^{\\infty}( G ) ; \ext{ f is invariant under the action of by right translations} } An interesting question is whether J_{\\mu} can retain enough information about to determine if \\mu has a nonzero continuous component . One of the main results of the present work is that this is never possible when G is a [SIN] group , where [SIN] denotes the 150%; MARGIN: 0cm 0cm 0pt" which admit arbitrarily small conjugation invariant neighbourhoods of the identity element . We prove that for every probability measure \\mu on a [SIN] group G there exists a discrete probability measure \u with J_{\\mu} = J_{\u} . We also consider some related questions for [SIN] groups . Given \\mu , does there exist an absolutely contin - uous or continuous probability measure \u with J_{\\mu} = J_{\u} ? Note that since J_{\\mu} = J_{\u} implies G_{\\mu} = G_{\u} , this question is not interesting without imposing some restrictions on , at least , G_{\\mu} . We show by example that when G_{\\mu} = G , there need not exist any absolutely continuous probability measure \u with J_{\\mu} = J_{\u} . Every ideal J_{\\mu} is contained in an ideal that is maximal in the set J(G) of all ideals of this form – and (when G \\in [SIN] ) we prove that for every maximal element I of J(G) there exists an absolutely continuous probability measure \\mu such that I = J_{\\mu} . In the course of these investigations some results about the ideals J_{\\mu} for general locally compact second countable groups are also obtained . For example , if N is a closed normal subgroup of G such that L_{0}^{1}( G , N ) \\subseteq J_{\\mu} for some \\mu , then N is necessarily amenable; when G contains a closed normal amenable subgroup such that G/N \\in [SIN] then for every ideal J_{\\mu} which is maximal in J(G) , there exist an absolutely continous probability measure \u' and a discrete probability measure \u" such that J_{\\mu} = J_{\u'} = J_{\u"} We also answer in the Question that if the set J_{d}(G) = { J_{\\mu} ; \\mu \\ : \ext{is a discrete probability measure on G } } has a largest element then G is necessarily amenable .