  Let $(\Omega, \mu)$ be a measure space and $\{\tau_\alpha\}_{\alpha\in
\Omega}$ be a normalized continuous Bessel family for a finite dimensional
Hilbert space $\mathcal{H}$ of dimension $d$. If the diagonal $\Delta\coloneqq
\{(\alpha, \alpha):\alpha \in \Omega\}$ is measurable in the measure space
$\Omega\times \Omega$, then we show that
  \begin{align*}
  \sup _{\alpha, \beta \in \Omega, \alpha\neq \beta}|\langle \tau_\alpha,
\tau_\beta\rangle |^{2m}\geq
\frac{1}{(\mu\times\mu)((\Omega\times\Omega)\setminus\Delta)}\left[\frac{
\mu(\Omega)^2}{{d+m-1 \choose m}}-(\mu\times\mu)(\Delta)\right], \quad \forall
m \in \mathbb{N}. \end{align*} This improves 47 years old celebrated result of
Welch [\textit{IEEE Transactions on Information Theory, 1974}]. We introduce
the notions of continuous cross correlation and frame potential of Bessel
family and give applications of continuous Welch bounds to these concepts. We
also introduce the notion of continuous Grassmannian frames.
