If $\varphi, g$ are positive and non-zero in $L^p(\mathbb R^n)$ and $L^q(\mathbb R^n)$ respectively, then so is their product in $L^1(\mathbb R^n)$
Solution 1:
Let $A$ be the set where $g$ is stricly positive. Since $\varphi$ as you said is strictly positive almost everywhere, there is a subset $B$ of $A$ with $\lambda(B) >0$ such that $g \cdot \varphi$ is strictly positive. Finally,
$$\int g \cdot\varphi d \lambda \geq \int_B g \cdot\varphi d \lambda >0.$$
Where you can check the last inequality in this link.