Question about general theory of areas and volumes

[Werg22]

This is an elementary question: restricting ourselves to the euclidean plane, is there a strict definition of what kind of set of points constitutes a region with area? For example, does a set of points describing a circle adjoined with an isolated point outside the circle still constitutes a figure with definite area? Now, in the entire space of euclidean geometry, is there a strict rule to decide what constitutes a solid with definite volume?

[Hurkyl]

This is an elementary question: restricting ourselves to the euclidean plane, is there a strict definition of what kind of set of points constitutes a region with area? For example, does a set of points describing a circle adjoined with an isolated point outside the circle still constitutes a figure with definite area?
Under most definitions of area, the area of that region is well defined and equal to zero. I assume you meant the disk enclosed by that circle, in which the area would be equal to the area of the disk. (because the area of the point is zero)

The general task of measuring subsets of the plane (or any topological space) is the subject of measure theory. For Euclidean spaces, the Lesbegue measure is most commonly used.

[DeaconJohn]

Werg22, Like Hurkyl intimated, your question is an ideal "jumping off point" for the study of Lebesgue measure theory and Lebesgue integration. It's one of those "innocent" looking questions in mathematics that turns out not to be so simple when one really gets into it. DJ