This　work　is　devoted　to　the　study　of　stochastic　differential　equations　(SDEs)　whose　diffusion　coefficient　sigma(s,　X.　(boolean　AND　s))　is　Lipschitz　continuous　with　respect　to　the　path　of　the　solution　process　X,　while　its　drift　coefficient　b(s,　X.　(boolean　AND　s),　QX(s))　is　only　measurable　with　respect　to　X　and　depends　continuously　(in　the　sense　of　the　1-Wasserstein　metric)　on　the　law　of　the　solution　process.　Embedded　in　a　mean-field　game,　the　weak　existence　for　such　SDEs　with　mean-field　term　was　recently　studied　by　Lacker　[Stochastic　Process.　Appl.,　125　(2015),　pp.　2856-2894]　and　Carmona　and　Lacker　[Ann.　Appl.　Probab.,　25　(2015),　pp.　1189-1231]　under　only　sequential　continuity　of　b(s,　X.　(boolean　AND　s),　QX(s))　in　Q(X)　with　respect　to　a　weak　topology,　but　for　uniqueness,　Carmona　and　Lacker　supposed　that　b　is　independent　of　the　mean-field　term.　We　prove　the　uniqueness　in　law　for　b　depending　on　the　mean-field,　and　the　proof　of　the　existence　of　a　weak　solution,　relatively　short　in　comparison　with　Carmona　and　Lacker＇s　work,　is　extended　in　section　5　of　this　paper　to　the　study　of　2-person　zero-sum　stochastic　differential　games　described　by　doubly　controlled　coupled　mean-field　forward-backward　SDEs　with　dynamics　whose　drift　coefficient　is　only　measurable　with　respect　to　the　state　process.