Modular metric spaces, I: Basic concepts

The notion of a modular is introduced as follows. A (metric) modular   on a set XX is a function w:(0,∞)×X×X→[0,∞]w:(0,∞)×X×X→[0,∞] satisfying, for all x,y,z∈Xx,y,z∈X, the following three properties: x=yx=y if and only if w(λ,x,y)=0w(λ,x,y)=0 for all λ>0λ>0; w(λ,x,y)=w(λ,y,x)w(λ,x,y)=w(λ,y,x) for all λ>0λ>0; w(λ+μ,x,y)≤w(λ,x,z)+w(μ,y,z)w(λ+μ,x,y)≤w(λ,x,z)+w(μ,y,z) for all λ,μ>0λ,μ>0. We show that, given x0∈Xx0∈X, the set Xw={x∈X:limλ→∞w(λ,x,x0)=0}Xw={x∈X:limλ→∞w(λ,x,x0)=0} is a metric space with metric dw∘(x,y)=inf{λ>0:w(λ,x,y)≤λ}, called a modular space  . The modular ww is said to be convex   if (λ,x,y)↦λw(λ,x,y)(λ,x,y)↦λw(λ,x,y) is also a modular on XX. In this case XwXw coincides with the set of all x∈Xx∈X such that w(λ,x,x0)<∞w(λ,x,x0)<∞ for some λ=λ(x)>0λ=λ(x)>0 and is metrizable by dw∗(x,y)=inf{λ>0:w(λ,x,y)≤1}. Moreover, if dw∘(x,y)<1 or dw∗(x,y)<1, then (dw∘(x,y))2≤dw∗(x,y)≤dw∘(x,y); otherwise, the reverse inequalities hold. We develop the theory of metric spaces, generated by modulars, and extend the results by H. Nakano, J. Musielak, W. Orlicz, Ph. Turpin and others for modulars on linear spaces.