Clear["Global`*"]
$Assumptions=σ>0 && xi∈Reals &&xj∈Reals &&μi∈Reals&&μj∈Reals && Pi ∈Reals  && dx∈Reals && dμ∈Reals &&dμ≠0
f[xi_,μi_]=FullSimplify[1/Sqrt[2Pi σ^2]Exp[-(xi-μi)^2/(2σ^2)]]
dU[xi_,μi_,μj_]=FullSimplify[(Log[f[xi,μi]]-Log[f[xi,μj]])/β]
gg[xi_,xj_,μi_,μj_]=Exp[-β FullSimplify[dU[xi,μi,μj]+dU[xj,μj,μi]]]
(* dx=xi-xj, dμ=μi-μj *)
g[dx_,dμ_]=gg[dx+xj,xj,μj+dμ,μj]
s = Simplify[Reduce[g[dx,dμ]>1,dx,Reals]]
M[dx_,dμ_]=Piecewise[{{1,s}},g[dx,dμ]]
(* x0 = xj - μj *)
fifj[dx_,dμ_,x0_]=FullSimplify[f[x0+μj+dx,μj+dμ] f[x0+μj,μj]]
I1[dx_,dμ_] =Integrate[fifj[dx,dμ,x0]M[dx,dμ],{x0,-Infinity,Infinity}]
v=FullSimplify[Integrate[I1[dx,dμ] M[dx,dμ],{dx,-Infinity,Infinity}]]
dμ=2σ
FullSimplify[v]
FullSimplify[N[v]]

((E^(-(((x0^2)+(((dx-dμ+x0))^2))/(2 (σ^2)))))/(2 π (σ^2)))1g