This article is devoted to the study of products of random operators of the form M_{0,n}=M_{0}..M_{n-1}, where (M_n) is an ergodic sequence of positive operators on the space of signed measures on a space X. Under suitable conditions, in particular, a Doeblin-type minoration suited for non conservative operators, we obtain asymptotic results of the form µ M_{0,n} \simeq µ(h)r_n π_n, where h is a random bounded function, (r_n) is a random non negative sequence and π_n is a random probability measure on X. Moreover, h, (r_n) and π_n do not depend on the choice of the measure µ. We prove additionally that n^{-1} log(r_n) converges almost surely to the Lyapunov exponent λ of the process (M_{0,n}) and that the sequence of random probability measures (π_n) converges weakly towards a random probability measure. These results are analogous to previous estimates from Hennion in the case of dxd matrices, that were obtained with different techniques, based on a projective contraction in Hilbert distance. In the case where the sequence (M_n) is i.i.d, we additionally exhibit an expression of the Lyapunov exponent λ as an integral with respect to the weak limit of the sequence of random probability measures pπnq and exhibit an oscillation behavior of (r_n) when λ=0. We provide a detailed comparison of our assumptions with the ones of Hennion and present some example of applications of our results, in particular in the field of population dynamics.