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Every -finite measure is a decomposable measure, the converse is not true. s-finite measures. A measure is called a s-finite measure if ...
Every free abelian group is torsion-free, but the converse is not true, as is shown by the additive group of the rational numbers Q. Even if A is not finitely generated, the size of its torsion-free part is uniquely determined, as is explained in more detail in the article on rank of an abelian group.
The converse may or may not be true, and even if true, the proof may be difficult. For example, the four-vertex theorem was proved in 1912, but its converse was proved only in 1997. [3] In practice, when determining the converse of a mathematical theorem, aspects of the antecedent may be taken as establishing context.
Lagrange's theorem raises the converse question as to whether every divisor of the order of a group is the order of some subgroup. This does not hold in general: given a finite group G and a divisor d of |G|, there does not necessarily exist a subgroup of G with order d.
The converse, however, is not true; [10] for example, the space of rationals, with the usual topology, is σ-compact but not hemicompact. The product of a finite number of σ-compact spaces is σ-compact. However the product of an infinite number of σ-compact spaces may fail to be σ-compact. [11]
The complexity function of a disjunctive sequence S over an alphabet of size k is p S (n) = k n. [ 1 ] Any normal sequence (a sequence in which each string of equal length appears with equal frequency) is disjunctive, but the converse is not true.