A general ratio ergodic theorem with weighted averages is shown by utilizing a method of R. V. Chacon. The theorem contains Chacon's general ergodic theorem as a special case. Let (X, Xk, m) be a a-finite measure space and let T be a linear contraction on L1(m). Let {wv,; n> 1} be a sequence of nonnegative numbers whose sum is one, and let {u,,; n>0} be the sequence defined by Un = WiUtt-1 + + 1ti'U0, 1U = 1 In this note we shall show the following THEOREM. If {p7,; n>0} is a sequence of nonnegative measurable functions vitih I Tgl O}PROOF. Let I be the positive integers, E all possible subsets, and ,u the measure on (f, 2) defined by ju({1})= 1 and lt({i})-= I--ufor i_ 2. Let {f; n > } be the sequence defined by fl 7 = I . . . W'--), 31=W,. Let S be the linear operator on L1(u) satisfying Sh1= , n0phn and Sh,=(1-3_11)hl for i>2, where hl,, denotes the indicator function of the set {n}. Then it is known (cf. [2]) that IISfl =I and Slth_(1)=u, for each n>0. Thus the direct product Sx Tof S and Tis a linear contraction on L1(M xm) and satisfies (Sx T)'thlJ(1, x)=S?Th/1( ] T1f(x)=unTnf(x). Now define a sequence {j3,,; n>0} of nonnegative measurable functions on (Ix X, Y> It, ux m) by P, (, x) = Snhl (i)p,, (x). I t is easily checked that Received by the editors December 3, 1971. AMS 1970 subject classcficatio;zs. Primary 47A35.