Measurable riesz spaces

We study measurable elements of a Riesz space E, i.e. elements e ∈ E \ {0} for which the Boolean algebra F of fragments of e is measurable. In particular, we prove that the set E of all measurable elements of a Riesz space E with the principal projection property together with zero is a σ-ideal of E. Another result asserts that, for a Riesz space E with the principal projection property the following assertions are equivalent. (1) The Boolean algebraU of bands of E is measurable. (2) E = E and E satisfies the countable chain condition. (3) E can be embedded as an order dense subspace of L (µ) for some probability measure µ.