Abstract
The subject of this chapter is the measurement of the areas of subsets of the plane R 2 = R ×R. The areas of elementary geometric figures, such as squares, rectangles, and triangles, are already known to us. By known to us we mean that, e.g., by defining the area of a rectangle to be the product of the lengths of its sides, we obtain quantities that agree with our intuition. Since every right-angle triangle is half a rectangle, the areas of right-angle triangles are also known to us. Similarly, we can obtain the area of a general triangle.
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Notes
- 1.
\(\bigcup _{n=1}^{\infty }A_{n} \equiv \bigcup \{ A_{n}: n \in \mathbf{N}\}\) and \(\bigcap _{n=1}^{\infty }A_{n} \equiv \bigcap \{ A_{n}: n \in \mathbf{N}\}\), see 2.1.
- 2.
The usual terminology is two-dimensional Lebesgue measure.
- 3.
⋃ k = 1 n A k , \(\bigcap _{k=1}^{n}A_{k}\) are defined by induction; see 2.1.
- 4.
This uses the axiom of countable choice.
- 5.
The axiom of countable choice may be avoided as in the proof of Theorem 2.1.3.
- 6.
In Caratheodory’s terminology [6], this says area is a metric outer measure.
- 7.
These group properties are basic to much of mathematics.
- 8.
The usual terminology is Lebesgue integral .
- 9.
This generalization is derived in 6.6.
- 10.
This is generalized to any continuous g in Theorem 6.6.4.
- 11.
(4.4.6) is actually valid under general conditions; see 6.6.
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Hijab, O. (2016). Integration. In: Introduction to Calculus and Classical Analysis. Undergraduate Texts in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-28400-2_4
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DOI: https://doi.org/10.1007/978-3-319-28400-2_4
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