let $g$ be a discrete group. a $g$-space $x$ is called a $g$-boundary if the action $g \curvearrowright x$ is minimal and strongly proximal. in this talk, we shall prove a continuous version of the well-studied intermediate factor theorem in the context of measurable dynamics. when a product group $g = \gamma_1 \times \gamma_2$ acts (by a product action) on the product of corresponding $\gamma_i$-boundaries $\partial \gamma_i$, we show that every intermediate factor $$x \times (\partial \gamma_1 \times \partial \gamma_2) \rightarrow y \rightarrow x$$ is a product (under some additional assumptions on $x$). we shall also compare it to its measurable analog proved by bader-shalom. this is a recent joint work with yongle jiang.