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Throughout, let $(R, m)$ be a Noetherian regular local ring of characteristic $p > 0$ and $I$ a nonzero ideal of $R$. Let $D(-) = Hom_R(-, E)$ be the Matlis dual functor, where $E = E_R(R/m)$ is the injective hull of the residue field $R/m$. The theory of $F$-modules was first recognized by Gennady Lyubeznik in 1997, [1]. Since then it has become a powerful tool which has significant applications to local cohomology and $D$-module theories and successfully used by many mathematicians. In this talk, after giving some basic definitions and theorems about $F$-module theory, we show that the support of the Matlis dual of any $F$-finite $F$-module $M$ with $0 \notin Ass_R(M)$ is the entire spectrum of $R$. This talk consists of results from the joint work with Gennady Lyubeznik, [2].