  This paper focuses on extensions of the classic Erd\H{o}s-Gallai Theorem for
the set of weighted function of each edge in a graph. The weighted function of
an edge $e$ of an $n$-vertex uniform hypergraph $\mathcal{H}$ is defined to a
special function with respect to the number of edges of the longest Berge path
containing $e$. We prove that the summation of the weighted function of all
edges is at most $n$ for an $n$-vertex uniform hypergraph $\mathcal{H}$ and
characterize all extremal hypergraphs that attain the value, which strengthens
and extends the hypergraph version of the classic Erd\H{o}s-Gallai Theorem.
