By Tarantello G.

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The m-dimensional Lebesgue measure is denoted by the same symbol λm as the ordinary volume. If the dimension is fixed, we sometimes omit the subscript and write simply λ, especially in the one-dimensional case. Hereafter in this section, the term “measure” refers to the Lebesgue measure. The σ -algebra of sets on which the m-dimensional Lebesgue measure is defined is denoted by Am ; sets from this σ -algebra are called Lebesgue measurable, or simply measurable. 1 Definition and Basic Properties of the Lebesgue Measure 43 As follows from the definition of the Carathéodory extension, for a measurable set A, λm (A) = inf λm (Pk ) Pk ∈ P m , k 1 Pk ⊃ A .

If the dimension is fixed, we sometimes omit the subscript and write simply λ, especially in the one-dimensional case. Hereafter in this section, the term “measure” refers to the Lebesgue measure. The σ -algebra of sets on which the m-dimensional Lebesgue measure is defined is denoted by Am ; sets from this σ -algebra are called Lebesgue measurable, or simply measurable. 1 Definition and Basic Properties of the Lebesgue Measure 43 As follows from the definition of the Carathéodory extension, for a measurable set A, λm (A) = inf λm (Pk ) Pk ∈ P m , k 1 Pk ⊃ A .

XN }, we have μ(A) = sup μ(E) | E ⊂ A, card(E) < +∞ . The reader can easily verify that μ is additive. (6) An example of a volume defined on the algebra of bounded sets and their complements (see Sect. 2, Example (1)) can be obtained as follows. Given a > 0, put μ(A) = 0 if A is bounded, a if A is unbounded. This volume will be useful for constructing various counterexamples. 3 We establish the basic properties of volume. Theorem Let μ be a volume on a semiring P, and let P , P , P1 , . . , Pn ∈ P.