  Let $(M,\mathcal A,\mu)$ be a probability space and $f:M\to M$ a
homeomorphism preserving the Borel ergodic probability measure $\mu$. Given
$\mathcal F$ a continuous one-dimensional $f$-invariant foliation of $M$ with
$C^1$ leaves, we show that if $f$ preserves a continuous $\mathcal{F}$-arc
length system, then we only have three possibilities for the conditional
measures of $\mu$ along $\mathcal F$, namely:
  - they are atomic for almost every leaf, or
  - for almost every leaf their support is a Cantor subset of the leaf or
  - for almost every leaf they are equivalent to the measure $\lambda_x$
induced by the invariant arc-length system over $\mathcal F$.
  This trichotomy classifies, for example, the possible disintegrations of
ergodic measures along foliations over which $f$ acts as an isometry, and also
disintegrations of ergodic measures along the center foliation preserved by
transitive partially hyperbolic diffeomorphisms with topological neutral center
direction.
