  It has been recently realized that for abundant dynamical systems on a
compact manifold, the set of points for which Lyapunov exponents fail to exist,
called the Lyapunov irregular set, has positive Lebesgue measure. In the
present paper, we show that under any physical noise, the Lyapunov irregular
set has zero Lebesgue measure and the number of such Lyapunov exponents is
finite. This result is a Lyapunov exponent version of Ara\'{u}jo's theorem on
the existence and finitude of time averages. Furthermore, we numerically
compute the Lyapunov exponents for a surface flow with an attracting
heteroclinic connection, which enjoys the Lyapunov irregular set of positive
Lebesgue measure, under a physical noise. This paper also contains the proof of
the disappearance of Lyapunov irregular behavior on a positive Lebesgue measure
set for a surface flow with an attracting homoclinic/heteroclinic connection
under a non-physical noise.
