A proof of Pesin's formula

Let f : M → M be a diffeomorphism of a compact manifold M and let χ: M → R be defined by putting χ( x ) equal to the sum of the non-negative characteristic exponents of f at x , each being counted with its multiplicity. If μ is an f -invariant probability of M which is absolutely continuous relative to Lebesgue measure, then Pesin has proved the entropy, h μ ( f ), is given by We prove this formula without using the theory of stable manifolds.