The Lebesgue-Feynman measure on a linear space E is a generalized measure on E which is defined as a linear functional μ on some linear space \(F(E,{\mathbb {C}})\) of functions \(E\to \mathbb {C}\) which is invariant with respect to the shift on any vector h ∈ E:

where S_h is the operator of the shift argument onto the vector h ∈ E, i.e. S_hu(x) = u(x + h),x ∈ E. There exists two different approach to definition of Lebesgue-Feynman measure. The first definition is given by Feynman formula and described in the work (Smolyanov and Shavgulidze 2015). In this paper we consider the second approach which is based on the definition of translation-invariant function of a set on some ring of elementary subsets. It will be interesting to investigate the relation between this two approaches to definition of the Lebesgue-Feynman measure.

上的线性空间中勒贝格-费曼量度é是在广义测度ê其被定义为线性的官能μ一些线性空间\（F（E，{\ mathbb {C}}）\）的功能\（E \到\ mathbb {C} \）是不变的相对于所述换档上的任何载体ħ ∈ Ë：

其中小号_ħ是移位参数的操作者到矢量ħ ∈ ë，即小号_ħ Ù（X）= Ü（X + ħ），X ∈ ë。Lebesgue-Feynman度量的定义有两种不同的方法。第一个定义由费曼公式给出，并在工作中有所描述（Smolyanov和Shavgulidze 2015）。在本文中，我们考虑第二种方法，该方法基于在基本子集的某些环上集合的平移不变函数的定义。研究这两种定义Lebesgue-Feynman测度的方法之间的关系将很有趣。