Summary: 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 have introduced \((p,q) \)-Bessel and \((p,q) \)-frame measures as a generalization of the concepts of Bessel and frame measures. In addition, we have defined the notions of \(q \)-Bessel sequence 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 have constructed a large number of examples of finite measures \(\mu\) which admit infinite \((p,q) \)-Bessel measures \(\nu \). We have showed 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 the convolutions of probability measures, one can obtain other \((p,q) \)-Bessel/frame measures for \(\mu \). We have presented a general way of constructing a \((p,q) \)-Bessel/frame measure for a given measure.