It is shown that the homogeneous ergodic bilinear averages with M\"{ o}bius or Liouville weight converge almost surely to zero, that is, if $T$ is a map acting on a probability space $(X,\mathcal{ A},\mu)$, and $a,b \in \mathbb{ Z}$, then for any $f,g \in L^2(X)$, for almost all $x \in X$, $${ \frac{ 1}{ N}}\sum_{ n=1}^{ N} \nu(n) f(T^{ an}x) g(T^{ bn}x) \longrightarrow 0, \text{ as } N \longrightarrow +\infty, $$ where $\nu$ is the Liouville function or the M\"{ o}bius function. We further obtain that the convergence almost everywhere holds for the short interval with the help of Zhan's estimation. Also our proof yields a simple proof of Bourgain's double recurrence theorem. Moreover, we establish that if $T$ is weakly mixing and its restriction to its Pinsker algebra has singular spectrum, then for any integer $k \geq 1$, for any $f_j\in L^{ \infty}(X)$, $j=1,\cdots,k,$ for almost all $x \in X$, we have $${ \frac{ 1}{ N}} \sum_{ n=1}^{ N} \nu(n) \prod_{ j=1}^{ k}f_j({ T_j}^nx) \longrightarrow 0, \text{ as } N \longrightarrow +\infty,$$ where $T_j$ are some powers of $T$, $j=1,\cdots,k$.
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