  A random chaotic interval map with noise which causes coarse-graining induces
a finite-state Markov chain. For a map topologically conjugate to a
piecewise-linear map with the Lebesgue measure being ergodic, we prove that the
Shannon entropy for the induced Markov chain possesses a finite limit as the
noise level tends to zero. In most cases, the limit turns out to be strictly
greater than the Lyapunov exponent of the original map without noise.
