Calculator Inputs
Example Data Table
| Case | Input | Method | Expected Measure |
|---|---|---|---|
| Overlapping intervals | [0, 3], [2, 5] | Merge to [0, 5] | 5 |
| Separated intervals | [1, 2], [4, 7] | Add 1 and 3 | 4 |
| Bounded complement | U=[0,10], A=[2,4]∪[7,8] | 10 − 3 | 7 |
| Rectangle product | [0,3] × [1,4] | 3 × 3 | 9 |
| Atomic measure | {1,2,2,5}, mass 0.5 | 3 unique atoms × 0.5 | 1.5 |
Formula Used
Interval union: μ(∪Iᵢ) = Σ(bⱼ − aⱼ), after overlapping intervals are merged.
Complement: μ(U \ A) = μ(U) − μ(A), when U is bounded and A is inside U.
Rectangle product: μ([x₁,x₂] × [y₁,y₂]) = |x₂ − x₁| × |y₂ − y₁|.
Atomic measure: μ(A) = Σwₓ, where each selected point has a chosen mass.
How to Use This Calculator
- Select the measure type that matches your set.
- Enter intervals, rectangles, or atomic points.
- Add a bounded universe when using complement mode.
- Set the scale factor if your measure is weighted.
- Choose decimal precision and a unit label.
- Press the calculate button.
- Review the result above the form.
- Download the result as CSV or PDF when needed.
Understanding Borel Measure
What This Tool Estimates
A Borel measure assigns size to Borel sets. These sets come from open intervals. They also include closed intervals. Countable unions and intersections are included too. In real analysis, the most familiar example is length. The calculator focuses on practical finite inputs. It handles unions of intervals. It handles bounded complements. It handles simple rectangles. It also supports atomic point mass.
Why Merging Matters
Overlap changes the final answer. The intervals [0,3] and [2,5] should not total six. Their union is [0,5]. Its measure is five. The calculator sorts intervals first. Then it merges touching or overlapping parts. This prevents double counting. Open and closed endpoints do not affect ordinary length. A single point has length zero.
Complement Calculations
A complement needs a reference universe. This tool uses a bounded interval for that universe. It clips excluded sets to that universe. Then it merges the clipped pieces. The excluded measure is subtracted from the universe length. This is useful in probability examples. It also helps with coverage problems. Always choose a universe that fits your model.
Rectangles and Atoms
Product measure gives area for rectangles. Each rectangle uses two horizontal endpoints. It also uses two vertical endpoints. The calculator multiplies side lengths. Listed rectangle areas are summed. Overlap is not removed in rectangle mode. Atomic mode is different. It gives positive mass to points. Duplicate points are counted once. This supports simple discrete Borel measures. Use the formula notes to check assumptions. Treat results as mathematical aids. They do not replace a formal proof.
FAQs
1. What is a Borel measure?
A Borel measure assigns a size to Borel sets. These sets are generated from open sets. On the real line, length measure is a common example.
2. Do open and closed endpoints change the answer?
For ordinary length measure, endpoints do not change the measure. A single endpoint has measure zero. Atomic mode is different because points can carry mass.
3. Why does the calculator merge intervals?
Merged intervals prevent double counting. If two intervals overlap, their shared part should be counted once. This gives the measure of the union.
4. Can I calculate complements?
Yes. Select complement mode. Enter a bounded universe and excluded intervals. The calculator subtracts the excluded union from the universe length.
5. Does rectangle mode remove overlapping area?
No. Rectangle mode sums each listed rectangle area. It does not subtract overlaps. Use it for disjoint rectangles or separate product measure checks.
6. What does the scale factor do?
The scale factor multiplies the calculated measure. Use it for weighted length, density scaling, or custom unit conversion. Use one for ordinary measure.
7. What is atomic point measure?
Atomic point measure gives mass to individual points. The calculator removes duplicate points and multiplies the distinct count by the mass per point.
8. Can I export my calculation?
Yes. After calculating, use the CSV or PDF buttons above the form. The files include the result, formula, and detail rows.