We establish a generalization of Bourgain double recurrence theorem and ergodic Bourgain-Sarnak's theorem by proving that for any aperiodic $1$-bounded multiplicative function $\boldsymbol{\nu}$, for any map $T$ acting on a probability space $(X,\A,\mu)$, for any integers $a,b$ , for any $f,g \in L^2(X)$, and for almost all $x \in X$, we have [ \frac{1}{N} \sum_{n=1}^{N}\boldsymbol{\nu}(n) f(T^{a n}x)g(T^{bn}x) \tend{N}{+\infty} 0. ]We further present with proof the key ingredients of Bourgain's proof of his double recurrence theorem.