Question

If \(\mu_{1}, \ldots, \mu_{n}\) are measures on \((X, \mathcal{M})\) and \(a_{1}, \ldots, a_{n} \in[0, \infty)\), then \(\sum_{1}^{n} a_{j} \mu_{j}\) is a measure on \((X, \mathcal{M})\).

Short answer

The sum \(\sum_{j=1}^{n} a_j \mu_j\) is a measure on \((X, \mathcal{M})\).

Step-by-step solution

Understanding Measures

A measure on a set \((X, \mathcal{M})\) is a function that assigns a non-negative value to each measurable subset in \(\mathcal{M}\), satisfying certain properties such as non-negativity, null empty set, and countable additivity.

Linear Combination of Measures

Given \(n\) measures \(\mu_1, \ldots, \mu_n\) on \((X, \mathcal{M})\) and non-negative numbers \(a_1, \ldots, a_n\), the expression \(\sum_{j=1}^{n} a_j \mu_j\) indicates a linear combination where each measure \(\mu_j\) is scaled by \(a_j\).

Properties of Non-negative Scalings

Each scaled measure \(a_j \mu_j\) is still a measure, since multiplying by a non-negative scalar preserves the measure properties: non-negativity, null empty set, and countable additivity.

Summation of Scaled Measures

The sum \(u = \sum_{j=1}^{n} a_j \mu_j\) is defined by \(u(E) = \sum_{j=1}^{n} a_j \mu_j(E)\) for each set \(E \in \mathcal{M}\). We need to verify if \(u\) satisfies the properties of a measure: non-negativity, null empty set, and countable additivity.

Verification of Properties

1. **Non-negativity:** Each \(\mu_j(E)\) is non-negative, and so is \(a_j \mu_j(E)\). Thus, \(u(E) \geq 0\).2. **Null empty set:** \(u(\emptyset) = \sum_{j=1}^{n} a_j \mu_j(\emptyset) = 0\) because \(\mu_j(\emptyset) = 0\) for all \(j\).3. **Countable additivity:** For a sequence of disjoint sets \((E_i)\), \(u(\bigcup E_i) = \sum_{j=1}^{n} a_j \mu_j(\bigcup E_i) = \sum_{j=1}^{n} a_j \sum \mu_j(E_i) = \sum \left(\sum_{j=1}^{n} a_j \mu_j(E_i) \right) = \sum u(E_i)\). This confirms \(u\) is \(\sigma\)-additive.

Conclusion

Since \(u(E) = \sum_{j=1}^{n} a_j \mu_j(E)\) satisfies all measure properties, \(u\) is a measure on \((X, \mathcal{M})\).

Key concepts

Non-negativity

In measure theory, non-negativity is one of the fundamental properties of measures. When we say a measure is non-negative, we mean that for any measurable set, the value assigned by the measure is always zero or positive.
This is similar to how volume works in geometry—after all, it wouldn't make sense to have a negative volume.
For any measurable subset \(E\) in the \(\sigma\)-algebra \(\mathcal{M}\), the measure \(\mu(E)\) must satisfy \(\mu(E) \geq 0\).

• This property ensures that we can meaningfully interpret measures as quantities representing sizes or probabilities, as negative sizes would defy these interpretations.
• Non-negativity also helps prevent contradictions within mathematical proofs involving measures.

In our exercise, each component of the linear combination of measures \(a_j \mu_j(E)\) is non-negative since both \(a_j\) and \(\mu_j(E)\) are non-negative.
Therefore, the sum \(u(E) = \sum_{j=1}^{n} a_j \mu_j(E)\) holds onto this non-negativity property and guarantees that the result is a measure.

Countable Additivity

Countable additivity, also known as \(\sigma\)-additivity, is another crucial property of measures. This property states that if you have a sequence of disjoint measurable sets \((E_i)\) in \(\mathcal{M}\), then the measure of the union of these sets can be calculated as the sum of their individual measures.

• In mathematical terms, given \(E_i\) are disjoint, \(\mu\left( \bigcup_{i=1}^{\infty} E_i \right) = \sum_{i=1}^{\infty} \mu(E_i)\).
• This concept is vital as it ensures that a measure can effectively handle infinite collections of sets just like finite ones.

In the context of the exercise, the linear combination \(u(E)\) respects countable additivity.
This is verified by seeing that if the individual measures \(\mu_j\) are \(\sigma\)-additive, then the sum \(u(E) = \sum_{j=1}^{n} a_j \mu_j(E)\) is also \(\sigma\)-additive, ensuring it's well-defined across infinite disjoint set collections.
This property makes measure theory tremendously powerful and applicable to fields such as probability, where handling infinite outcomes is often required.

Linear Combination of Measures

The idea of a linear combination is quite straightforward—it's about adding things together, but with a twist. Instead of just adding them, each item is multiplied by a constant before you sum it up.
In the context of measures, given \(n\) measures \(\mu_1, \ldots, \mu_n\) on our set \((X, \mathcal{M})\) and non-negative numbers \(a_1, \ldots, a_n\), the expression \(\sum_{j=1}^{n} a_j \mu_j\) defines their linear combination.

• The terms \(a_j\) are like weights that dictate the influence of each measure \(\mu_j\) in the combination.
• Each \(a_j \mu_j\) is still a measure because scaling by a non-negative constant preserves the core properties of measures.

The significance of forming a linear combination of measures is that it enables us to construct new measures from known ones.
By appropriately choosing coefficients \(a_j\), we can emphasize or de-emphasize certain measures according to our needs, ultimately resulting in a valid measure \(u\).
This flexibility is particularly useful in applications where different criteria must be balanced judiciously, such as in statistical mechanics or when forming statistical estimates.