Lebesgue Measure Calculator

Calculation type

Use inf or -inf for unbounded limits.

Display precision

Precision only affects display, not union merging.

Quick examples

Examples replace the current inputs.

Intervals (one per line)

Format: [a,b], (a,b], [a,b), or (a,b). Overlaps are merged automatically for union length.

Example data table

Set description	Representation	Lebesgue measure
Unit interval	[0,1]	1
Two disjoint intervals	[0,1] ∪ [3,5]	3
Countable set	ℤ	0
Box in R²	[0,2] × [-1,1]	4
Unbounded ray	[0, inf)	∞

The calculator supports merging overlaps for unions, giving the union’s true length in one dimension.

Formula used

1D interval: μ([a,b]) = b − a (same for (a,b), [a,b), (a,b]).
Union in R: μ(∪ Iᵢ) equals the length of the merged disjoint union.
Countable set: μ(A) = 0 for any countable A ⊂ Rⁿ.
Box in Rⁿ: μ(∏[aᵢ,bᵢ]) = ∏(bᵢ − aᵢ).
Unbounded: any interval or box with infinite side length has μ = ∞.

How to use this calculator

Select a calculation type that matches your set.
Enter bounds using numbers, inf, or -inf.
For unions, place one interval on each line.
Press Calculate measure to see results above the form.
Use CSV or PDF export buttons in the result panel.

Why measure matters for real problems

Lebesgue measure turns “size” into a reliable number that behaves well under limits and unions. In one dimension it matches ordinary length, but it also supports sets built from many pieces. This calculator focuses on practical, measurable inputs: finite unions of intervals, axis-aligned boxes, and known zero-measure families. It is designed for teaching, checking homework, and quick engineering estimates.

Intervals: length without endpoint anxiety

For any real bounds a ≤ b, an interval has measure b − a. Whether endpoints are open or closed does not change length, so [0,1], (0,1], and (0,1) all measure 1. Degenerate intervals like [5,5] contribute 0, which is useful when validating data imports. If you type decimals, the calculator keeps precision internally, then formats the display to your chosen decimals.

Union of intervals: overlap-safe totals

Summing lengths naively can double-count overlaps. The calculator merges overlaps and touching pieces first, then adds the disjoint lengths. For example, [-1,2] ∪ (1,4] becomes a single block [-1,4] with measure 5. If you enter [0,1] and [1,3], the union still measures 3 because the shared point does not change length. The merged-interval list is shown so you can audit what was combined.

Boxes in Rⁿ: product structure

For a box ∏[aᵢ,bᵢ], the measure equals ∏(bᵢ − aᵢ). A 3D box [0,1]×[-2,2]×[3,6] has side lengths 1, 4, and 3, so volume is 12. The graph shows side lengths per axis, helping spot a zero side that forces total measure to 0. This is a common safeguard in geometry pipelines.

Zero-measure sets you meet often

Any finite or countable subset of Rⁿ has Lebesgue measure 0. That includes integers Z, rationals Q, and any explicitly listed finite sample. This matters in probability and integration: changing a function on a countable set does not change its Lebesgue integral. In simulations, it explains why isolated spikes rarely affect area-based metrics.

Unbounded inputs and infinite results

If any interval has an infinite endpoint, or any box side is unbounded, the measure is infinite. Use “inf” and “-inf” to model rays like [0, inf). The calculator still exports results, and the chart keeps bounded components visible while signaling unbounded parts. When ∞ appears, treat it as a qualitative conclusion: the set is not finite in size, so comparisons should use normalization or truncation.

FAQs

1) Does an open interval have a different measure than a closed interval?

No. For real bounds a < b, (a,b), [a,b], (a,b], and [a,b) all have measure b − a. Endpoints do not affect length.

2) Why does the union mode merge intervals first?

Merging prevents double-counting overlap. If two intervals intersect or touch, their union length is smaller than the sum of individual lengths. The calculator computes the correct union size.

3) What happens if I enter “inf” or “-inf” as a bound?

The calculator treats those as unbounded limits. Any interval or box side with infinite length produces an overall measure of ∞.

4) Why is the measure of the integers or rationals equal to zero?

Any countable set in Rⁿ has Lebesgue measure 0. Integers and rationals can be listed in a sequence, so they are countable and therefore measure zero.

5) Can this compute the measure of a Cantor-type set?

Not directly. The tool targets intervals, unions, and boxes. For fractal sets, you usually need a construction rule and limit process rather than a finite input list.

6) Are exported CSV and PDF values exact?

Exports use the computed numeric result and your set description. Display precision controls formatting, but merging and arithmetic are performed consistently before formatting.