Explore measure completion through precise set descriptions. Check values, complements, densities, and null-cover consistency instantly. Build stronger intuition for measurable extensions and completed spaces.
Use the form below for a finite-space numerical model of measure completion. The page checks whether a candidate set can be treated as a completed measurable set.
For a measure space (X, Σ, μ), the completion adds every set that can be written as:
E = A ∪ N, where A ∈ Σ and N ⊆ Z for some Z ∈ Σ with μ(Z) = 0.
The completed measure is defined by:
μ̄(E) = μ(A)
For a finite total space, the calculator also reports:
μ̄(X \ E) = μ(X) - μ̄(E)
Density = [μ̄(E) / μ(X)] × 100
Null-cover ratio = [μ(Z) / μ(X)] × 100
The subset estimate inside the null cover is not a formal measure-theory requirement. It is a practical numerical check that helps confirm the entered representation is internally consistent.
These sample rows show how the completion rule preserves the measurable core value.
| Case | μ(A) | μ(Z) | Subset Estimate | μ(X) | Displayed Measure | μ̄(E) | Complement | Density | Status |
|---|---|---|---|---|---|---|---|---|---|
| Exact null cover | 12.5000 | 0.0000 | 0.0000 | 40.0000 | 12.5000 | 12.5000 | 27.5000 | 31.25% | Valid completion model |
| Tiny numerical null | 8.7500 | 0.0000004 | 0.0000002 | 25.0000 | 8.7500001 | 8.7500 | 16.2500 | 35.00% | Valid within tolerance |
| Review needed | 15.0000 | 0.2000 | 0.1800 | 30.0000 | 15.1800 | 15.0000 | 15.0000 | 50.00% | Needs review |
It evaluates a numerical model of measure completion. You enter a measurable core, a null measurable cover, and total space information. The page then reports the completed measure, complement, density, and consistency checks.
In the completion, the extra piece N sits inside a measurable null set Z. Because null additions do not change measure, the completed measure of E = A ∪ N remains the measure of the measurable core A.
Real numerical workflows often contain rounding error. The tolerance lets a very small null-cover value behave like zero, which is useful when importing approximate data from simulations, spreadsheets, or measurement logs.
No. The page does not prove measurability from arbitrary descriptions. It checks whether your numerical representation matches the completion template, assuming the theoretical conditions behind the entered sets are already justified.
That field lets you compare an external value with the completed result. It helps identify rounding drift, imported-data differences, or misunderstandings about whether null contributions were incorrectly added to the core measure.
The status changes when the null cover exceeds tolerance, the subset estimate is too large for the declared null cover, or the combined values no longer fit inside the chosen total space.
Yes. Probability measures are measures with total mass one. Enter the measurable core probability, a null event cover, and the total value 1 to study completed events and their complements.
The page can download the computed summary as CSV or PDF after calculation. It can also export the example table as CSV, which is useful for teaching notes, documentation, and audits.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.