Positive projections and conditional mathematical expectations

The non-negative projections in $L^1$ space are considered. A non-negative projection in $L^1$ is a linear operator $T\colon L^1\to L^1$ such that $T^2=T$ and $T\geq0$. In this paper we give a description of such projections in term of conditional expectation operators. Another authors considered the case of positive projection on $L^1$ which is also contractive whereas we do not require this condition. It is proved that every non-negative projection in $L^1$ space is “nearly” a conditional expectation operator.