关于L^2(Ω,F,P)空间上投影映射与条件数学期望的等价性研究

A research on equivalence between projection mapping and conditional expectation in L^2(Ω,F,P) space

目的讨论了L2(Ω,F,P)空间上投影映射与条件数学期望的等价性。方法采用了逻辑推理的方法进行了证明。结果证明了L2(Ω,F,P)空间上的投影映射就是条件数学期望E(ξ|R)。结论表明在L2(Ω,F,P)空间上,条件数学期望E(ξ|R)是唯一满足投影方程的投影映射。

Aim The equivalence of the projection mapping and the conditional expectation in L^2（Ω,F,P） space are investigated. Methods The equivalence is proved with logical inference meth- od. Results It is proved that the projection mapping in L^2（Ω,F,P） space is conditional expectation E（ξ｜R）. Conclusion The results show that the conditional expectation is the only projection mapping that meets the projection equation.