Now assume F(<G() for all X, and prove stochastic dominance follows l(x)d/(x)= ∫(x)dG(x)+∫(x)(dF(x)-dG(x)= ∫(x)dG(x)+∫(x)(dH(x)-dG(x) Let H(x)=F(x)-G(x), so we need to know if Ju(x)dH(x)>0 for all functions u(x) Integrate by parts to get ∫(x)dH(x)=[l(x)(x)8-∫a(x)H(xtx H(O)=0 and limx+ H(x)=0 so [u(x)H(x)10 equals zero. The second term is negative if H(X<0 everywhere, so were doneNow assume F  x  G  x for all x, and prove stochastic dominance follows.  uxdFx   uxdGx   uxdFx  dGx   uxdGx   uxdFx  dGx Let H(x)F(x)-G(x), so we need to know if  uxdHx  0 for all functions u(x). Integrate by parts to get  uxdHx  uxHx0    u xHxdx H(0)0 and limx Hx  0 so uxHx0  equals zero.The second term is negative if H(x) 0 everywhere, so we’re done