  We study multifractal properties in time evolution of a single particle
subject to repeated measurements. For quantum systems, we consider circuit
models consisting of local unitary gates and local projective measurements. For
classical systems, we consider models for estimating the trajectory of a
particle evolved under local transition processes by partially measuring
particle occupations. In both cases, multifractal behaviors appear in the
ensemble of wave functions or probability distributions conditioned on
measurement outcomes after a sufficiently long time. While the nature of
particle transport (diffusive or ballistic) qualitatively affects the
multifractal properties, they are even quantitatively robust to the measurement
rate or specific protocols. On the other hand, multifractality is generically
lost by generalized measurements allowing erroneous outcomes or by
postselection of the outcomes with no particle detection. We demonstrate these
properties by numerical simulations and also propose several simplified models,
which allow us to analytically obtain multifractal properties in the monitored
single-particle systems.
