The time evolution and the asymptotic outcome of a Landau-Zener-Stueckelberg-Majorana (LZ) process under continuous weak non-selective measurement is analyzed. We compare two measurement protocols in which the populations of either the adiabatic or the non-adiabatic levels are (continuously and weakly) monitored. The weak measurement formalism, described using a Gaussian Kraus operator, leads to a time evolution characterized by a Markovian dephasing process, which, in the non-adiabatic measurement protocol is similar to earlier studies of LZ dynamics in a dephasing environment. Casting the problem in the language of measurement theory makes it possible for us to compare diabatic and adiabatic measurement scenarios, to consider engineered dephasing as a control device and to examine the manifestation of the Zeno effect under the different measurement protocols. In particular, under measurement of the non- adiabatic populations, the Zeno effect is manifested not as a freezing of the measured system in its initial state, but rather as an approach to equal asymptotic populations of the two diabatic states. This behavior can be traced to the way by which the weak measurement formalism behaves in the strong measurement limit, with a built-in relationship between measurement time and strength.
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