Expected Value of a sequence with two random variables
The &quot;periodicity of the cosine&quot; has little to do with the matter at hand. Indeed, unless $\omega/\pi$ is a rational number, it is not even the case that the discrete-time sequence $x[n] = A\cos(n\omega + \phi)$ is a periodic sequence at all. See, for example, this question which asks &quot;Why is $\cos\left(\frac n6\right)$ aperiodic?&quot; Now, coming to the meat of the question and correcting meaning-less notation such as $\omega \in [\omega-\lambda,\omega+\lambda]$ to $\omega \in [\omega_0-\lambda,\omega_0+\lambda]$, we have from LOTUS, the law of the unconscious statistician, that
$$E\left[x[n]\right] = E[A\cos(n\omega + \phi)] = A 
\int_{-\pi}^{\pi}\int_{\omega_0-\lambda}^{\omega_0+\lambda}\cos(n\omega + \phi)f_{\omega,\phi}(\omega,\phi)\, \mathrm d\omega \,\mathrm d\phi \tag{1}$$ where $f_{\omega,\phi}(\omega,\phi)$ is the joint density of $\omega $ and $\phi$. The OP has not stated that $\omega$ and $\phi$ are independent random variables, but if they are assumed to be independent (as is most likely the intention), then the joint density in $(1)$ factors into the product of the marginal densities (which are given to be uniform on $[\omega_0-\lambda,\omega_0+\lambda]$ and $[-\pi,\pi]$ respectively,) and so $(1)$ can be written as $$E\left[x[n]\right] =  A 
\int_{-\pi}^{\pi}\frac{1}{2\pi}\left[\frac{1}{2\lambda}\int_{\omega_0-\lambda}^{\omega_0+\lambda} \cos(n\omega + \phi) \, \mathrm d\omega\right] \,\mathrm d\phi. \tag{2}$$
Similarly, the autocorrelation function is given by
$$E\left[x[n]x[m]\right] =  A 
\int_{-\pi}^{\pi}\frac{1}{2\pi}\left[\frac{1}{2\lambda}\int_{\omega_0-\lambda}^{\omega_0+\lambda} \cos(n\omega + \phi)\cos(m\omega + \phi) \, \mathrm d\omega\right] \,\mathrm d\phi. \tag{2}$$