Class LebesgueIntegration

java.lang.Object

org.episteme.core.mathematics.analysis.integration.LebesgueIntegration

public class LebesgueIntegration
extends Object

Lebesgue integration support.

Focuses on methods that handle complex domains and high dimensions, characteristic of Lebesgue integration utility.

Since:

1.0

Author:

Silvere Martin-Michiellot, Gemini AI (Google DeepMind)

Constructor Summary

Constructors

Constructor	Description
LebesgueIntegration()

Method Summary

Modifier and Type	Method	Description
static Real	integrateSimpleFunction(List<Real> values, List<Real> measures)	Integration of a simple function (step function).
static Real	monteCarlo(Function<Vector<Real>, Real> f, Vector<Real> lowerBounds, Vector<Real> upperBounds, int samples)	Monte Carlo integration.

Methods inherited from class Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Constructor Details

LebesgueIntegration

public LebesgueIntegration()

Method Details

monteCarlo

public static Real monteCarlo(Function<Vector<Real>, Real> f,
 Vector<Real> lowerBounds,
 Vector<Real> upperBounds,
 int samples)

Monte Carlo integration.

Approximates the integral of f over a hypercube defined by bounds. Converges as O(1/√N), independent of dimension.

Parameters:

f - function to integrate (takes Vector, returns Real)

lowerBounds - lower bounds of the hypercube

upperBounds - upper bounds of the hypercube

samples - number of random samples

Returns:

approximate integral

integrateSimpleFunction

public static Real integrateSimpleFunction(List<Real> values,
 List<Real> measures)

Integration of a simple function (step function).

Fundamental to definition of Lebesgue integral. ∫ f dμ = Σ a_i * μ(E_i)

Parameters:

values - values (a_i) taken by the function

measures - measures μ(E_i) of the sets where function takes value a_i

Returns:

the integral value