Question

Given a measure space \((X, \mathcal{M}, \mu)\) and \(E \in \mathcal{M}\), define \(\mu_{E}(\mathcal{A})=\mu(A \cap E)\) for \(A \in \mathcal{M}\). Then \(\mu_{E}\) is a measure.

Short answer

\(\mu_E\) is a measure since it satisfies all measure properties.

Step-by-step solution

Understand the Measure Definition

A measure is a function \( \mu : \mathcal{M} \to [0, \infty] \) that satisfies three properties: non-negativity, null empty set, and \(\sigma\)-additivity.

Check Non-Negativity

For any \( A \in \mathcal{M} \), \( \mu_E(A) = \mu(A \cap E) \geq 0 \) since \( \mu \) is non-negative and measures intersections as subsets of measurable sets.

Verify the Null Empty Set Property

Consider \( A = \emptyset \). Then \( A \cap E = \emptyset \), and thus \( \mu_E(\emptyset) = \mu(\emptyset \cap E) = \mu(\emptyset) = 0 \). This satisfies the property for measures.

Establish \(\sigma\)-additivity

Take a countable collection \( \{A_i\}_{i=1}^{\infty} \subseteq \mathcal{M} \). We have \( \mu_E\left(\bigcup_{i=1}^{\infty} A_i\right) = \mu\left(\bigcup_{i=1}^{\infty} (A_i \cap E)\right) \). Since \(\mu\) is \(\sigma\)-additive, \( \mu\left(\bigcup_{i=1}^{\infty} (A_i \cap E)\right) = \sum_{i=1}^{\infty} \mu(A_i \cap E) \), hence \( \mu_E\left(\bigcup_{i=1}^{\infty} A_i\right) = \sum_{i=1}^{\infty} \mu_E(A_i) \).

Key concepts

Non-Negativity

In measure theory, one of the foundational properties of a measure is non-negativity. This means that for any measurable set \( A \), the measure \( \mu(A) \) must be greater than or equal to zero. This makes sense because a measure can be thought of as assigning a size or volume to sets, and it's logical that size can't be negative.

When checking non-negativity for our specific measure \( \mu_E \), defined as \( \mu_E(A) = \mu(A \cap E) \), we need to consider the intersection \( A \cap E \).
Since \( \mu \) is already a non-negative measure, the intersection \( A \cap E \) as a subset is also measured in a non-negative way:

• For any \( A \in \mathcal{M} \), \( \mu_E(A) = \mu(A \cap E) \geq 0 \).

This confirms that non-negativity holds for \( \mu_E \) just as it does for the original measure \( \mu \).
Understanding this is crucial because it ensures all measurements are logically consistent; you cannot have a negative size in a system measuring size or probability.

Null Empty Set

The null empty set property is another important attribute of a measure. This property essentially states that the measure of an empty set is always zero.
\(\mu(\emptyset) = 0 \) signifies that there's nothing to measure, hence it cannot possess any volume or size.

When we look at \( \mu_E \), the measure is slightly adapted but retains this fundamental property.
Consider:

• The measure \( \mu_E(\emptyset) = \mu(\emptyset \cap E) \).
• Since \( \emptyset \cap E = \emptyset \), then \( \mu_E(\emptyset) = \mu(\emptyset) = 0 \).

This shows that whether you measure directly through \( \mu \) or indirectly through \( \mu_E \), empty sets remain unmeasurable in the sense of having a measure of zero.
Grasping this concept helps underline that non-existent elements in a set don't mistakenly contribute to the total measure, maintaining both theoretical and practical precision.

Sigma-additivity

Sigma-additivity (or \( \sigma \)-additivity) is the property that ensures the measure can handle countable unions of sets in a consistent manner.
This is an essential trait as it ensures that a measure can combine over an infinite count of sets while retaining reliability and accuracy.

To verify \( \sigma \)-additivity for \( \mu_E \), consider a countable collection of sets \( \{A_i\}_{i=1}^{\infty} \):

• With \( \mu_E \), we define: \( \mu_E\left(\bigcup_{i=1}^{\infty} A_i\right) = \mu\left(\bigcup_{i=1}^{\infty} (A_i \cap E)\right) \).
• Because the original measure \( \mu \) is \( \sigma \)-additive, it applies: \( \mu\left(\bigcup_{i=1}^{\infty} (A_i \cap E)\right) = \sum_{i=1}^{\infty} \mu(A_i \cap E) \).
• Thus, \( \mu_E \) follows that \( \mu_E\left(\bigcup_{i=1}^{\infty} A_i\right) = \sum_{i=1}^{\infty} \mu_E(A_i) \).

Sigma-additivity showcases the robustness of measure theory, allowing it to manage infinite scenarios in a fluid, coherent manner.
By doing so, we ensure our measure remains reliable whether dealing with simple or complex and infinite set arrangements.