I'm considering two function $f,g\in L^{1}(\Omega)$ which satisfy\begin{align} \int_{\Omega}f dx\leq\int_{\Omega}g dx\end{align}Now, if I consider more a function $\phi\in C^{1}(\Omega)$ such that $0\leq\phi\leq1$ in $\Omega$. Will the inequality\begin{align} \int_{\Omega}\phi f dx\leq \int_{\Omega}\phi gdx \end{align}still holds true?
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