  Entanglement in quantum many-body systems can exhibit universal phenomena
governed by long-distance properties. We study universality and phase
transitions of the entanglement inherent to open many-body systems, namely, the
entanglement between a system of interest and its environment. Specifically, we
consider the Tomonaga-Luttinger liquid (TLL) under a local measurement and
analyze its unconditioned nonunitary evolution, where the measurement outcomes
are averaged over. We quantify the system-environment entanglement by the
R\'enyi entropy of the post-measurement density matrix, whose size-independent
term encodes the universal low-energy physics. We develop a field-theoretical
description to relate the universal term to the $g$ function in a boundary
conformal field theory (CFT), and use the renormalization group (RG) method and
the boundary CFT techniques to determine its value. We show that the universal
contribution is determined by the TLL parameter $K$ and can exhibit singularity
signifying an entanglement phase transition. Surprisingly, in certain cases the
size-independent contribution can increase as a function of the measurement
strength in contrast to what is na\"ively expected from the $g$-theorem. We
argue that this unconventional behavior could be attributed to the dangerously
irrelevant term which has been found in studies of the resistively shunted
Josephson junction. We also check these results by numerical calculations in
the spin-$\frac{1}{2}$ XXZ chain subject to a site-resolved measurement.
Possible experimental realization in ultracold gases, which requires no
postselections, is discussed.
