  In this paper, we prove measurability of event for which a general
continuous-time stochastic process satisfies continuous-time Metric Temporal
Logic (MTL) formula. Continuous-time MTL can define temporal constrains for
physical system in natural way. Then there are several researches that deal
with probability of continuous MTL semantics for stochastic processes. However,
proving measurability for such events is by no means an obvious task, even
though it is essential. The difficulty comes from the semantics of "until
operator", which is defined by logical sum of uncountably many propositions.
Given the difficulty involved in proving the measurability of such an event
using classical measure-theoretic methods, we employ a theorem from stochastic
analysis. This theorem is utilized to prove the measurability of hitting times
for stochastic processes, and it stands as a profound result within the theory
of capacity. Next, we provide an example that illustrates the failure of
probability approximation when discretizing the continuous semantics of MTL
formulas with respect to time. Additionally, we prove that the probability of
the discretized semantics converges to that of the continuous semantics when we
impose restrictions on diamond operators to prevent nesting.
