  We extend a result of Ledrappier, Hochman, and Solomyak on exact
dimensionality of stationary measures for $\text{SL}_2(\mathbb{R})$ to
disintegrations of stationary measures for $\text{GL}(\mathbb{R}^d)$ onto the
one dimensional foliations of the space of flags obtained by forgetting a
single subspace.
  The dimensions of these conditional measures are expressed in terms of the
gap between consecutive Lyapunov exponents, and a certain entropy associated to
the group action on the one dimensional foliation they are defined on. It is
shown that the entropies thus defined are also related to simplicity of the
Lyapunov spectrum for the given measure on $\text{GL}(\mathbb{R}^d)$.
