  Ledrappier and Young introduced a relation between entropy, Lyapunov
exponents and dimension for invariant measures of diffeomorphisms on compact
manifolds. In this paper, we show that a self-affine measure on the plane
satisfies the Ledrappier-Young formula if the corresponding iterated function
system (IFS) satisfies the strong separation condition and the linear parts
satisfy the dominated splitting condition. We give sufficient conditions,
inspired by Ledrappier and by Falconer and Kempton, that the dimensions of such
a self-affine measure is equal to the Lyapunov dimension. We show some
applications, namely, we give another proof for Hueter-Lalley's theorem and we
consider self-affine measures and sets generated by lower triangular matrices.
