  We start with the task of discriminating finitely many multipartite quantum
states using LOCC protocols, with the goal to optimize the probability of
correctly identifying the state. We provide two different methods to show that
finitely many measurement outcomes in every step are sufficient for approaching
the optimal probability of discrimination. In the first method, each
measurement of an optimal LOCC protocol, applied to a $d_{\rm loc}$-dim local
system, is replaced by one with at most $2d_{\rm loc}^2$ outcomes, without
changing the probability of success. In the second method, we decompose any
LOCC protocol into a convex combination of a number of "slim protocols" in
which each measurement applied to a $d_{\rm loc}$-dim local system has at most
$d_{\rm loc}^2$ outcomes. To maximize any convex functions in LOCC (including
the probability of state discrimination or fidelity of state transformation),
an optimal protocol can be replaced by the best slim protocol in the convex
decomposition without using shared randomness. For either method, the bound on
the number of outcomes per measurement is independent of the global dimension,
the number of parties, the depth of the protocol, how deep the measurement is
located, and applies to LOCC protocols with infinite rounds, and the
"measurement compression" can be done "top-down" -- independent of later
operations in the LOCC protocol. The second method can be generalized to
implement LOCC instruments with finitely many outcomes: if the instrument has
$n$ coarse-grained final measurement outcomes, global input dimension $D_0$ and
global output dimension $D_i$ for $i=1,...,n$ conditioned on the $i$-th
outcome, then one can obtain the instrument as a convex combination of no more
than $R=\sum_{i=1}^n D_0^2 D_i^2 - D_0^2 + 1$ slim protocols; in other words,
$\log_2 R$ bits of shared randomess suffice.
