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The Lebesgue measure of an interval, of a box, of $Q$ and of the irrationals in $[0, 1]$

Example

Assume the Axiom of Countable Choice. Then every bounded interval in $R$ has Lebesgue measure equal to its length, every box in $R^{n}$ has Lebesgue measure equal to the product of its side lengths, $λ_{1} (Q) = 0$ , and

$λ_{1} ([0, 1] ∖ Q) = 1.$

In particular a degenerate interval and the empty box both have measure $0$ .

Facts & Assumptions

Given: The Axiom of Countable Choice.

[L1]

Assuming countable choice, a box in $R^{n}$ with parameters $a_{i} \leq b_{i}$ is Lebesgue measurable of measure $\prod_{i < n} (b_{i} - a_{i})$ (A box in $R^{n}$ with parameters $a_{i} \leq b_{i}$ is Lebesgue measurable of measure $\prod_{i < n} (b_{i} - a_{i})$ , whichever of its faces are included).

[L2]

Every at most countable subset of $R^{n}$ is Lebesgue null; in particular $λ_{1} (Q) = 0$ (Every at most countable subset of $R^{n}$ is Lebesgue null; in particular $λ_{1} (Q) = 0$ ).

[L3]

Assuming countable choice, $λ_{n}$ is a complete measure (Assuming countable choice, $L (R^{n})$ is a sigma-algebra containing every elementary set and $λ_{n}$ is a complete measure extending elementary volume).

Verification

technique · direct

1.1L1

By [L1], every interval with endpoints $a \leq b$ and every box with real side parameters has Lebesgue measure equal to its geometric length or volume, whatever choice of open and closed faces is made.

1.2L2

The rationals form a Lebesgue null subset of $R$ , so $λ_{1} (Q) = 0$ .

2.1step 1.1step 1.2L3algebra

Since $[0, 1] = (Q \cap [0, 1]) ⊔ ([0, 1] ∖ Q)$ and $λ_{1} ([0, 1]) = 1$ by step 1.1, step 1.2 and [L3] give $λ_{1} ([0, 1] ∖ Q) = 1$ .

3.1step 1.1∎

Step 1.1 also covers the boundary cases: if $a = b$ then the interval has measure $0$ , and the empty box has measure $0$ .

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

32 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

John K. Hunter, Measure Theory (UC Davis lecture notes), Chapter 2 (standard reference, not scraped)
T. Tao, An Introduction to Measure Theory (GSM 126), Section 1.2 (standard reference, not scraped)

The Lebesgue measure of an interval, of a box, of QQ and of the irrationals in [0,1][0,1]