Given two continuous functions $$f,g \colon I \to\mathbb{R}$$ f , g : I → R such that g is positive and f/g is strictly monotone, a measurable space $$(T,\mathcal{A})$$ ( T , A ) , a measurable family of d -variable means $$m: I^{d} \times T \to I$$ m : I d × T → I , and a probability measure μ on the measurable sets $$\mathcal{A}$$ A , the d -variable mean $$M_{f,g,m;\mu} \colon I^{d} \to I$$ M f , g , m ; μ : I d → I is defined by $$M_{f,g,m;\mu}({\bf x}) :=\Bigl(\frac{f}{g}\Bigr)^{-1}\biggl( \frac{\int_{T} f(m(x_{1},\ldots,x_{d},t)){\rm d}\mu(t)} {\int_{T} g(m(x_{1},\ldots,x_{d},t)){\rm d}\mu(t)}\biggr) \quad ({\bf x} =(x_{1},\ldots,x_{d})\in I^{d}).$$ M f , g , m ; μ ( x ) : = ( f g ) - 1 ( ∫ T f ( m ( x 1 , … , x d , t ) ) d μ ( t ) ∫ T g ( m ( x 1 , … , x d , t ) ) d μ ( t ) ) ( x = ( x 1 , … , x d ) ∈ I d ) . The aim of this paper is to solve the equality and homogeneity problems of these means, i.e., to find conditions for the generating functions ( f, g ) and ( h, k ), for the family of means m , and for the measure μ such that the equality $$M_{f,g,m;\mu}( {\bf x} )=M_{h,k,m;\mu}( {\bf x} ) \quad ( {\bf x} \in I^{d})$$ M f , g , m ; μ ( x ) = M h , k , m ; μ ( x ) ( x ∈ I d ) and the homogeneity property $$M_{f,g,m;\mu}(\lambda {\bf x} ) = \lambda M_{f,g,m;\mu}( {\bf x} ) \quad (\lambda>0,\, {\bf x} ,\lambda {\bf x} \in I^{d}), $$ M f , g , m ; μ ( λ x ) = λ M f , g , m ; μ ( x ) ( λ > 0 , x , λ x ∈ I d ) , respectively, be satisfied.

中文翻译：

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广义积分均值的相等性和齐次性

给定两个连续函数 $$f,g \colon I \to\mathbb{R}$$ f , g : I → R 使得 g 为正且 f/g 严格单调，一个可测空间 $$(T,\ mathcal{A})$$ ( T , A ) ，一个可测量的 d 族变量意味着 $$m: I^{d} \times T \to I$$ m : I d × T → I ，以及一个概率在可测量集 $$\mathcal{A}$$ A 上测量 μ，d 变量均值 $$M_{f,g,m;\mu} \colon I^{d} \to I$$ M f ， , 米 ; μ : I d → I 由 $$M_{f,g,m;\mu}({\bf x}) :=\Bigl(\frac{f}{g}\Bigr)^{-1} 定义\biggl( \frac{\int_{T} f(m(x_{1},\ldots,x_{d},t)){\rm d}\mu(t)} {\int_{T} g( m(x_{1},\ldots,x_{d},t)){\rm d}\mu(t)}\biggr) \quad ({\bf x} =(x_{1},\ldots, x_{d})\in I^{d}).$$ M f , g , m ; μ ( x ) : = ( fg ) - 1 ( ∫ T f ( m ( x 1 , … , xd , t ) ) d μ ( t ) ∫ T g ( m ( x 1 , … , xd , t ) ) d μ ( t ) ) ( x = ( x 1 , … , xd ) ∈ I d ) 。本文的目的是解决这些均值的相等性和同质性问题，即找到生成函数 ( f, g ) 和 ( h, k )、均值族 m 和测度 μ 的条件使得等式 $$M_{f,g,m;\mu}( {\bf x} )=M_{h,k,m;\mu}( {\bf x} ) \quad ( {\bf x} } \in I^{d})$$ M f , g , m ; μ ( x ) = M h , k , m ; μ ( x ) ( x ∈ I d ) 和同质性 $$M_{f,g,m;\mu}(\lambda {\bf x} ) = \lambda M_{f,g,m;\mu} ( {\bf x} ) \quad (\lambda>0,\, {\bf x} ,\lambda {\bf x} \in I^{d}), $$ M f , g , m ; μ ( λ x ) = λ M f , g , m ; μ ( x ) ( λ > 0 , x , λ x ∈ I d ) 分别满足。g,m;\mu}( {\bf x} )=M_{h,k,m;\mu}( {\bf x} ) \quad ( {\bf x} \in I^{d})$ $M f , g , m ; μ ( x ) = M h , k , m ; μ ( x ) ( x ∈ I d ) 和同质性 $$M_{f,g,m;\mu}(\lambda {\bf x} ) = \lambda M_{f,g,m;\mu} ( {\bf x} ) \quad (\lambda>0,\, {\bf x} ,\lambda {\bf x} \in I^{d}), $$ M f , g , m ; μ ( λ x ) = λ M f , g , m ; μ ( x ) ( λ > 0 , x , λ x ∈ I d ) 分别满足。g,m;\mu}( {\bf x} )=M_{h,k,m;\mu}( {\bf x} ) \quad ( {\bf x} \in I^{d})$ $M f , g , m ; μ ( x ) = M h , k , m ; μ ( x ) ( x ∈ I d ) 和同质性 $$M_{f,g,m;\mu}(\lambda {\bf x} ) = \lambda M_{f,g,m;\mu} ( {\bf x} ) \quad (\lambda>0,\, {\bf x} ,\lambda {\bf x} \in I^{d}), $$ M f , g , m ; μ ( λ x ) = λ M f , g , m ; μ ( x ) ( λ > 0 , x , λ x ∈ I d ) 分别满足。