Example Data Table
| Case |
X intervals |
Points |
Density |
Atom mass |
Expected result |
| Overlapping union |
[0,2], (1,4] |
None |
1 |
0 |
Merged interval is [0,4], measure is 4. |
| Interval plus atom |
(0,1), [3,5] |
2 |
1 |
0.25 |
Total estimate is 3.25. |
| Window complement |
[1,3] |
None |
1 |
0 |
Inside [0,5], complement length is 3. |
| Product set |
[0,2] |
None |
1 |
0 |
With Y=[1,4], product measure is 6. |
Formula Used
For a finite interval, the base measure is:
μ([a,b]) = b - a
For a finite union, intervals are sorted and merged first:
μ(∪ Iᵢ) = Σ length(Jᵢ)
Here, each Jᵢ is a disjoint merged interval. Endpoint type does not change length. Weighted interval measure is:
weighted measure = density × merged length
Point atoms are added separately:
total estimate = density × merged length + Σ atom masses
The bounded complement is:
density × (window length - length of set inside window)
The affine image under x ↦ ax + b uses:
transformed interval measure = |a| × interval measure
For a product set A × B, the estimate is:
μ(A × B) = μ(A) × μ(B)
How To Use This Calculator
- Enter one interval per line, such as [0,2] or (1,4].
- Enter points as numbers. Use x:mass for a custom point mass.
- Set the density scale for interval length.
- Enter a window to calculate a bounded complement.
- Enter an affine multiplier to estimate scaled images.
- Add Y intervals when checking product set measure.
- Press Calculate to view the result above the form.
- Use CSV or PDF export for a saved report.
Understanding Borel Measure
A Borel measure calculator helps you study measurable sets on the real line. It focuses on finite descriptions. You enter intervals, points, bounds, and weights. The tool then reduces the set into disjoint pieces. This makes the result easier to check.
What The Calculator Measures
In common use, the Borel measure of an interval equals its length. Open and closed endpoints do not change that length. A single point has zero Lebesgue measure. Some measures also add atomic mass to listed points. This calculator supports both ideas. It can treat intervals by length. It can also add a selected mass for each point.
Why Merging Matters
Many Borel sets are written as unions. These unions may overlap. For example, [0,3] and (2,5] share part of the line. Counting both lengths separately would overstate the measure. The calculator sorts all intervals. It then merges touching or overlapping pieces. The final length comes from these merged intervals only.
Advanced Options
The window fields estimate a relative complement. Enter a bounded reference interval. The calculator compares your set inside that window. It returns the measure outside the set but inside the window. The affine multiplier estimates the measure after the map x becomes ax plus b. Length scales by the absolute value of a.
Product Sets
A product set uses two one dimensional sets. If A and B are finite interval unions, the rectangle product A × B has measure equal to measure(A) times measure(B). This option is helpful for simple plane regions. It gives a clean result without drawing every rectangle.
Practical Use
Use exact numbers when possible. Separate each interval on a new line. Use brackets to show endpoint style. Remember that endpoint style usually changes topology, not length. Review the merged interval list before trusting the final number. It reveals overlaps, gaps, and duplicated pieces. Export the result when you need a record for notes, homework, or reports.
Limits To Remember
No finite form can describe every Borel set. Some sets need countable operations. Others need proofs. This tool handles practical finite inputs. It gives a transparent numerical model. Use it as a guide, not as a replacement for careful formal verification in analysis.
FAQs
What is a Borel measure?
A Borel measure assigns sizes to Borel sets. These sets are generated from open sets through countable unions, intersections, and complements.
Can this calculate every Borel set?
No. It handles finite interval unions, points, complements inside a bounded window, affine scaling, and simple product sets.
Do open endpoints change the measure?
For ordinary interval length, no. The intervals [0,1], (0,1), [0,1), and (0,1] all have length 1.
Why are intervals merged first?
Merging prevents double counting. If two intervals overlap, the shared part should only contribute once to the union measure.
What does atom mass mean?
Atom mass gives a selected point a positive measure. This lets the calculator model simple measures beyond ordinary length.
How should I enter point masses?
Enter plain points like 3 or 5. Use 3:0.25 when a point needs its own mass instead of the default value.
What is the complement window?
It is a bounded reference interval. The calculator measures the part of that window not covered by your interval union.
What does product measure estimate?
It estimates the measure of A × B by multiplying the calculated X set measure with the calculated Y set measure.