里斯表示定理附加性质探讨

Study on the Additional Properties of Riesz Representation Theorem

里斯表示定理是连接泛函分析与测度论的桥梁,本文通过例子说明在该定理的假设下得到的Borel测度一般不具有正则性,只有加强该定理的条件,才可得到相应的正则测度。

The Riesz representation theorem is a bridge between the functional analysis and the measure theory.In the general case,one can not prove the regularity of the resulting Borel measure under the usual assumptions of Riesz representation theorem.In this paper,we give an explicit example to illustrate this situation.However,when some condition of Riesz representation theorem is strengthened,the corresponding regular Borel measure can be obtained.