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A set function mu is said to possess countable subadditivity if, given any countable disjoint collection of sets {E_k}_(k=1)^n on which mu is defined, mu( union ...

The Euclidean metric is the function d:R^n×R^n->R that assigns to any two vectors in Euclidean n-space x=(x_1,...,x_n) and y=(y_1,...,y_n) the number ...

A Cantor set with Lebesgue measure greater than 0.

Let f(theta) be Lebesgue integrable and let f(r,theta)=1/(2pi)int_(-pi)^pif(t)(1-r^2)/(1-2rcos(t-theta)+r^2)dt (1) be the corresponding Poisson integral. Then almost ...

A set function mu is finitely additive if, given any finite disjoint collection of sets {E_k}_(k=1)^n on which mu is defined, mu( union _(k=1)^nE_k)=sum_(k=1)^nmu(E_k).

The French metro metric is an example for disproving apparently intuitive but false properties of metric spaces. The metric consists of a distance function on the plane such ...

If f^'(x) is continuous and the integral converges, int_0^infty(f(ax)-f(bx))/xdx=[f(0)-f(infty)]ln(b/a).

For n>=3, there exist no additive finite and invariant measures for the group of displacements in R^n.

The term "homology group" usually means a singular homology group, which is an Abelian group which partially counts the number of holes in a topological space. In particular, ...

A measure algebra which has many properties associated with the convolution measure algebra of a group, but no algebraic structure is assumed for the underlying space.