  Considering a finite Borel measure $ \mu $ on $ \mathbb{R}^d $, a pair of
conjugate exponents $ p, q $, and a compatible semi-inner product on $ L^p(\mu)
$, we introduce $ (p,q) $-Bessel and $ (p,q) $-frame measures as a
generalization of the concepts of Bessel and frame measures. In addition, we
define notions of $ q $-Bessel and $ q$-frame in the semi-inner product space $
L^p(\mu) $. Every finite Borel measure $\nu$ is a $(p,q)$-Bessel measure for a
finite measure $ \mu $. We construct a large number of examples of finite
measures $ \mu $ which admit infinite $ (p,q) $-Bessel measures $ \nu $. We
show that if $ \nu $ is a $ (p,q) $-Bessel/frame measure for $ \mu $, then $
\nu $ is $ \sigma $-finite and it is not unique. In fact, by using convolutions
of probability measures, one can obtain other $ (p,q) $-Bessel/frame measures
for $ \mu $. We present a general way of constructing a $ (p,q) $-Bessel/frame
measure for a given measure.
