Question

If \(f \in L^{+}\), let \(\lambda(E)=\int_{E} f d \mu\) for \(E \in \mathcal{M}\). Then \(\lambda\) is a measure on \(\mathcal{M}\), and for any \(g \in L^{+}, \int g d \lambda=\int f g d \mu\). (First suppose that \(g\) is simple.)

Short answer

For a simple function \(g\), \(\int g \, d\lambda = \int fg \, d\mu\).

Step-by-step solution

Define Simple Function

A function \(g\) is called simple if it can be expressed as \(g = \sum_{i=1}^{n} a_i \chi_{E_i}\), where \(a_i\) are non-negative constants and \(E_i\) are disjoint measurable sets, and \(\chi_{E_i}\) is the characteristic function of the set \(E_i\).

Calculate Integral with Measure \(\lambda\)

For a simple function \(g = \sum_{i=1}^{n} a_i \chi_{E_i}\), calculate the integral with respect to \(\lambda\): \[\int g \, d\lambda = \int \left( \sum_{i=1}^{n} a_i \chi_{E_i} \right) d\lambda = \sum_{i=1}^{n} a_i \lambda(E_i) = \sum_{i=1}^{n} a_i \int_{E_i} f \, d\mu.\]

Calculate Integral with Measure \(\mu\)

Calculate the integral of \(fg\) with respect to the measure \(\mu\):\[\int fg \, d\mu = \int \left( \sum_{i=1}^{n} a_i f \chi_{E_i} \right) d\mu = \sum_{i=1}^{n} a_i \int_{E_i} f \, d\mu.\]

Compare Integrals

Compare \(\int g \, d\lambda\) with \(\int fg \, d\mu\):Both are equal as \[\int g \, d\lambda = \sum_{i=1}^{n} a_i \int_{E_i} f \, d\mu = \int fg \, d\mu.\] This equality holds because they are calculated in the same way for simple functions.

Key concepts

Understanding Simple Functions

Simple functions play a fundamental role in measure theory and integration. They serve as stepping stones for understanding more complex integrals. A simple function is one that can be described as a finite sum of scaled characteristic functions. To express a simple function, we use the formula \(g = \sum_{i=1}^{n} a_i \chi_{E_i}\), where each \(a_i\) is a non-negative constant, \(E_i\) are disjoint, measurable sets, and \(\chi_{E_i}\) represents the characteristic function of the set \(E_i\).
The characteristic function \(\chi_{E_i}\) is defined as being 1 if a point belongs to the set \(E_i\), and 0 otherwise. This feature allows simple functions to

1. be easily manipulated,
2. provide approximations for more complex functions, and
3. aid in computing integrals.

Simple functions form the backbone of the Lebesgue integration process because any non-negative measurable function can be approximated as closely as desired by simple functions.

Lebesgue Integral: A New Way to Integrate

The Lebesgue integral is a powerful tool used in real analysis and measure theory to integrate more general functions than the classical Riemann integral. Unlike the Riemann integral which partitions the domain of a function, the Lebesgue integral partitions the range, allowing it to handle functions with many discontinuities.
In the context of simple functions, suppose you have a simple function \(g = \sum_{i=1}^{n} a_i \chi_{E_i}\). The Lebesgue integral of \(g\) with respect to a measure \(\lambda\) can be expressed as \(\int g \, d\lambda = \sum_{i=1}^{n} a_i \lambda(E_i)\). Here, each term \(a_i \lambda(E_i)\) represents the contribution of the set \(E_i\), scaled by the constant \(a_i\).
This method efficiently integrates functions, whether they appear simple or complex. The elegance of the Lebesgue integral lies in its flexibility to incorporate varying function values across different segments of their domain without being sensitive to their discontinuities.

Grasping Measure Space

A measure space is a foundational concept in measure theory, providing the structure to measure sizes of different sets. It is defined as a triplet \((X, \mathcal{M}, \mu)\), where:

• \(X\) is the set under consideration,
• \(\mathcal{M}\) is a \(\sigma\)-algebra of subsets of \(X\), and
• \(\mu\) is a measure function that assigns a non-negative value to each set in \(\mathcal{M}\), reflecting its size or probability.
  

The measure \(\mu\) captures the idea of 'size' in a mathematical sense, extending beyond lengths and areas to include more abstract 'sizes' of sets, crucial in fields like probability and real analysis.
In our exercise, \(\lambda(E) = \int_{E} f \, d\mu\) represents a measure defined through integration, asserting how an integral can create a new measure, \(\lambda\), based on an existing measure, \(\mu\). This concept is key in forming an understanding of how different measures can interact in integration and analysis.