  In this paper, we generalize the upper bound in Varadhan's Lemma. The
standard formulation of Varadhan's Lemma contains two important elements,
namely an upper semicontinuous integrand and a rate function with compact
sublevel sets. However, motivated by results from queueing theory, we do not
assume that rate functions have compact sublevel sets. Moreover, we drop the
assumption that the integrand is upper semicontinuous and replace it by a
weaker condition. We prove that the upper bound in Varadhan's Lemma still holds
under these weaker conditions. Additionally, we show that only measurability of
the integrand is required when the rate function is continuous.
