Citation Check: Indicator Integrals and Lebesgue Measure
The highlighted passage claims that, once \(E\) is assumed to be Lebesgue measurable, the identity
\[ \int \mathbf{1}_E\,d\lambda=\lambda(E) \]
is not an extra theorem requiring a difficult proof, but belongs to the basic definitional machinery of the Lebesgue integral. This is mathematically right in the usual construction of the Lebesgue integral: one first defines the integral of nonnegative simple functions, and an indicator function \(\mathbf{1}_E\) is the simplest such function. For a measurable set \(E\), the definition gives
\[ \int \mathbf{1}_E\,d\lambda = 1\cdot \lambda(E) = \lambda(E), \]
with the value allowed to be \(+\infty\) if \(E\) has infinite Lebesgue measure.
The citation question is narrower: do the cited references, especially Folland and Royden–Fitzpatrick, support the highlighted sentence? No retrieved source passages were supplied, so the exact wording and page-level confirmation cannot be checked here. Still, from standard knowledge of these textbooks, the fit is strong. Folland’s Real Analysis and Royden–Fitzpatrick’s Real Analysis both construct Lebesgue integration from measurable simple functions, and in that construction characteristic functions of measurable sets have integrals equal to the measures of those sets [Folland 1999; Royden and Fitzpatrick 2010]. The highlighted sentence is therefore supported in substance.
There are two small citation-quality issues. First, the supplied reference list includes Kolmogorov, but the highlighted passage cites Folland and Royden–Fitzpatrick, not Kolmogorov. Kolmogorov is relevant to the later probability version \(\mathbb{E}[\mathbf{1}_E]=P(E)\), but it is not the most direct source for the Lebesgue-measure claim in the highlighted passage [Kolmogorov 1956]. Second, because no page, section, theorem, or retrieved quotation was provided, the citation cannot be verified at the level of an exact textual match. The mathematically appropriate next step is to inspect the sections in Folland and Royden–Fitzpatrick where the integral of nonnegative simple functions is defined.
The passage’s phrase “part of the definition and basic structure” is also appropriate but slightly interpretive. In many treatments, the equality for \(\mathbf{1}_E\) is literally immediate from the definition of the simple-function integral. In other presentations, it may appear as an early proposition or consequence of that definition. Either way, the cited textbooks should support the weaker and stronger readings: the formula is at least an immediate basic consequence of the construction, and often essentially built into the definition.
Citation mapping
| Document quote | Reference quote and location | Judgement comments |
|---|---|---|
| Under the measurable-set assumption | No retrieved quote available. Inspect Folland, Real Analysis, 2nd ed., chapter/section on measurable functions and integration of nonnegative simple functions; inspect Royden–Fitzpatrick, Real Analysis, 4th ed., chapters on Lebesgue measurable functions and Lebesgue integration. | Supported in substance by standard measure-theoretic definitions: \(\mathbf{1}_E\) is measurable exactly when \(E\) is measurable, relative to the relevant \(\sigma\)-algebra. Exact textbook wording cannot be confirmed without page/section text. |
| the equation \(\int \mathbf{1}_E\,d\lambda=\lambda(E)\) | No retrieved quote available. Best source location to check: the definition of the integral of a nonnegative simple measurable function in Folland 1999 and Royden–Fitzpatrick 2010. | Strongly supported in substance. In the usual definition, if \(s=\sum_i a_i\mathbf{1}_{E_i}\) with measurable \(E_i\), then \(\int s\,d\mu=\sum_i a_i\mu(E_i)\). Taking \(s=\mathbf{1}_E\), \(a_1=1\), and \(\mu=\lambda\) gives the displayed formula. |
| is part of the definition and basic structure of the Lebesgue integral | No retrieved quote available. Inspect the initial construction of the Lebesgue integral for simple functions in Folland 1999 and Royden–Fitzpatrick 2010. | Fairly supported, but the wording is interpretive. Depending on the textbook presentation, the formula may be stated directly in the definition of simple-function integration or follow immediately as a first consequence. The citation supports the mathematical content even if not necessarily this exact phrasing. |
| The context claim: if \(E\) is not measurable, then \(\lambda(E)\) is not defined in the usual Lebesgue theory and \(\mathbf{1}_E\) is not Lebesgue measurable. | No retrieved quote available. Inspect the definitions of Lebesgue measurable sets, Lebesgue measure as a measure on the Lebesgue \(\sigma\)-algebra, and measurable functions in Folland 1999 or Royden–Fitzpatrick 2010. | Supported under the standard convention that \(\lambda\) denotes Lebesgue measure on the \(\sigma\)-algebra of Lebesgue measurable sets. Nuance: some texts define Lebesgue outer measure \(\lambda^*\) on all subsets of \(\mathbb{R}\), but \(\lambda(E)\) as a measure is normally reserved for measurable \(E\). |
| Supplied reference includes Kolmogorov 1956. | No retrieved quote available. Kolmogorov’s Foundations of the Theory of Probability concerns probability spaces and expectations, not specifically the construction of the Lebesgue integral with Lebesgue measure. | Relevant to the later probability identity \(\mathbb{E}[\mathbf{1}_E]=P(E)\), but not the best source for the highlighted Lebesgue-integral construction. It is not needed to support this highlighted sentence. |
Confidence level: Medium-high (80%).
Why confidence is below 100%
The mathematical claim is standard and very likely supported by Folland 1999 and Royden–Fitzpatrick 2010. Confidence is not 100% because no retrieved source text, page number, theorem number, definition number, or section locator was supplied. Therefore I cannot verify the exact wording or the precise location in either textbook. The citation is substantively correct, but a final citation audit should inspect the original sections where each book defines the integral of nonnegative simple functions and discusses characteristic functions of measurable sets.
References
Folland, Gerald B. Real Analysis: Modern Techniques and Their Applications. 2nd ed., Wiley, 1999. Cited source to check for the construction of the Lebesgue integral from simple functions.
Kolmogorov, A. N. Foundations of the Theory of Probability. 2nd English ed., Chelsea Publishing Company, 1956. Supplied reference; relevant to probability foundations, but only indirectly relevant to the highlighted Lebesgue-integral claim.
Royden, H. L., and P. M. Fitzpatrick. Real Analysis. 4th ed., Pearson, 2010. Cited source to check for the Lebesgue integral and the integral of characteristic functions.