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homology
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Quick question: what does it mean for a measure to have finite mass? (is this another way of saying sigma finite or something?)
Thanks,
Kevin
Thanks,
Kevin
Lonewolf said:I wonder if you've come across linear operators called m-currents recently? Is this a question from geometric measure theory?
A measure with finite mass is a mathematical concept used in the field of measure theory. It refers to a type of measure that assigns a finite value to sets within a given space. This measure is often used to quantify the size or extent of a set or collection of objects.
A measure with finite mass differs from other types of measures, such as countable measures or infinite measures, in that it assigns a finite value to sets instead of an infinite or countable value. This makes it useful for measuring finite quantities or objects.
Examples of measures with finite mass include the Lebesgue measure, which is used to measure the size of sets in Euclidean space, and the counting measure, which assigns a value of 1 to each element in a set. Other examples include the length, area, and volume measures used in geometry and calculus.
Measures with finite mass have a wide range of applications in various fields, such as physics, economics, and statistics. They can be used to measure quantities, probabilities, and distributions, and are often used in modeling and analysis of real-world phenomena.
Some key properties of measures with finite mass include additivity, which means that the measure of a union of disjoint sets is equal to the sum of their individual measures, and monotonicity, which means that the measure of a larger set is greater than the measure of a smaller set. Measures with finite mass also satisfy countable subadditivity and translation invariance properties.