  We consider quantum many-body dynamics under quantum measurements, where the
so-called measurement-induced phase transitions (MIPs) occur, which depend on
the frequency of the measurement. In this work, we consider the robustness of
the MIP for long-range interaction that decays as $r^{-\alpha}$ with distance
$r$. The effects of long-range interactions are classified into two regimes: i)
for $\alpha > \alpha_c$, the MIP is observed and the universality of the
exponent depends on $\alpha$; ii) for $\alpha<\alpha_c$, the MIP is absent,
even for arbitrarily strong measurements. Using fermion models, we demonstrate
both regimes in integrable and non-integrable cases. In addition, we identify
the underlying mechanism and propose sufficient conditions to observe the MIP,
that is, $\alpha > d+1/2$ for general bilinear systems and $\alpha > d+1$ for
nonintegrable general systems ($d$: spatial dimension).
