Calculator Inputs
Enter each measurable piece as a value-measure pair. For best interpretation, use disjoint sets or a clean partition.
Example Data Table
This sample treats the function as a finite simple function over four measurable pieces.
| Set Label | Function Value | Measure | Contribution |
|---|---|---|---|
| A1 | 2 | 1 | 2 |
| A2 | -1 | 0.5 | -0.5 |
| A3 | 0.5 | 2 | 1 |
| A4 | 3 | 1.5 | 4.5 |
| Total Lebesgue Integral | 7 | ||
Formula Used
Simple function form: f = Σ ai 1Ei
Lebesgue integral: ∫ f dμ = Σ aiμ(Ei)
Positive part: ∫ f+ dμ = Σ max(ai, 0)μ(Ei)
Negative part magnitude: ∫ f- dμ = Σ max(-ai, 0)μ(Ei)
L¹ norm: ‖f‖1 = Σ |ai|μ(Ei)
This calculator evaluates a finite simple function or a discretized measurable partition. It is ideal for piecewise-constant data, signed contributions, and measure-weighted summaries.
How to Use This Calculator
- Enter one row for each measurable piece or partition segment.
- Type the function value assigned to that measurable set.
- Enter the nonnegative measure for the same set.
- Adjust decimal precision and graph order if needed.
- Click Compute Integral to display the result below the header.
- Review the graph, contribution table, positive part, and negative magnitude.
- Use Download CSV or Download PDF to save the analysis.
FAQs
1. What does this calculator actually compute?
It computes the Lebesgue integral for a finite simple function or discretized measurable model. Each row contributes value multiplied by measure, then all rows are summed.
2. Can I enter negative function values?
Yes. Negative values are allowed. The calculator separates positive and negative parts, then combines them into the signed Lebesgue integral.
3. Why must the measure stay nonnegative?
A measure assigns nonnegative size to measurable sets. Negative measures would break the intended interpretation of weighted set size in standard Lebesgue integration.
4. Does this tool symbolically integrate arbitrary formulas?
No. It is designed for simple-function style input and discretized partitions. For arbitrary symbolic formulas, a symbolic analysis engine would be needed first.
5. What is the difference between the integral and the L¹ norm?
The integral keeps signs, so positive and negative parts can cancel. The L¹ norm uses absolute values, so every contribution increases the total magnitude.
6. When is the average value meaningful?
It is meaningful when total measure is positive. The calculator divides the integral by total measure, giving a measure-weighted mean value.
7. Should the measurable pieces be disjoint?
Yes, preferably. Disjoint pieces make the interpretation clean because each function value is assigned to a separate measurable set without overlap.
8. What does the graph show?
The bars show each row’s contribution to the integral. The line shows how the cumulative integral changes as you move through the measurable pieces.