  In this paper, we will define a signed Lattice measure on $\sigma$-algebras,
as well as give the definition of positive and negative Lattice. Herein, we
will show that the Hahn Decomposition Theorem decomposes any space X into a
positive lattice A and a negative Lattice B such that $A \vee B$ =X and the
signed Lattice measure of $A \wedge B $ is 0.
