Question

Suppose \(\left\\{\nu_{j}\right\\}\) is a sequence of positive measures. If \(\nu_{j} \perp \mu\) for all \(j\), then \(\sum_{1}^{\infty} \nu_{j} \perp \mu ;\) and if \(\nu_{j} \ll \mu\) for all \(j\), then \(\sum_{1}^{\infty} \nu_{j} \ll \mu\).

Short answer

If \( \nu_j \perp \mu \) or \( \nu_j \ll \mu \) holds for all \( j \), the sum maintains the same relation to \( \mu \).

Step-by-step solution

Understanding Measure Notation

In measure theory, \( u_{j} \perp \mu \) means \( u_{j} \) and \( \mu \) are mutually singular measures, implying there exists a set \( A \) such that \( u_{j}(A) = 0 \) and \( \mu((X\setminus A)) = 0 \), where \( X \) is the entire space. Conversely, \( u_{j} \ll \mu \) means \( u_{j} \) is absolutely continuous with respect to \( \mu \), indicating that if \( \mu(B) = 0 \) for some set \( B \), then \( u_{j}(B) = 0 \).

Assess Mutual Singularity

Given \( u_{j} \perp \mu \) for all \( j \), each measure \( u_{j} \) is singular with respect to \( \mu \). There exist measurable sets \( A_j \) such that \( u_{j}(A_j) = 0 \) and \( \mu(X \setminus A_j) = 0 \). The mutual singularity implies that \( \sum_{j=1}^{\infty} u_{j} \perp \mu \) because the union of sets \( A_j \) can be used to find a larger set \( A \) such that \( \sum u_j(A) = 0 \) and \( \mu(X \setminus A) = 0 \).

Assess Absolute Continuity

Given \( u_{j} \ll \mu \) for all \( j \), each \( u_{j} \) is absolutely continuous with respect to \( \mu \). This means for any measurable set \( B \) where \( \mu(B) = 0 \), it follows directly that \( u_{j}(B) = 0 \). Thus, the sum \( \sum_{j=1}^{\infty} u_{j} \) is also absolutely continuous with respect to \( \mu \) because absolute continuity is preserved under countable sums.

Key concepts

Mutual Singularity

In measure theory, mutual singularity is an intriguing concept that helps us understand how two measures relate to each other. When we say that two measures \(u_j\) and \(\mu\) are mutually singular, denoted as \(u_j \perp \mu\), it implies that the measures live on separate parts of the space. Specifically, there exists a set \(A\) such that \(u_j(A) = 0\) and \(\mu(X \setminus A) = 0\), where \(X\) is the universal set under consideration.
This means the measure \(u_j\) has no 'mass' on the set where \(\mu\) measures, and vice-versa. In practical terms, if you visualize \(X\) as a pie, \(u_j\) and \(\mu\) never overlap on any slice of this pie. Each has its own distinct slice or collection of slices where it takes values, but where one has values, the other has none.

• Mutual singularity implies separation. No common overlap exists in their supports.
• This concept extends nicely when dealing with countable sequences of measures, preserving singularities across sums.

Absolute Continuity

Absolute continuity is another fundamental concept in measure theory, essential for understanding how one measure behaves in relation to another. If we say \(u_j \ll \mu\), then \(u_j\) is absolutely continuous with respect to \(\mu\). This relationship signifies that wherever \(\mu\) assigns zero measure, \(u_j\) must also assign zero measure.
In more intuitive terms, absolute continuity ensures that \(u_j\) 'echoes' the behavior of \(\mu\) over space. Suppose \(\mu(B) = 0\) for some set \(B\). Then, absolute continuity will ensure that \(u_j(B) = 0\) as well. \(u_j\) cannot assign any measure where \(\mu\) sees none.

• Easily extended to sequences. If all measures in a sequence \(u_j\) are absolutely continuous with respect to \(\mu\), their sum is too.
• This helps in analyzing changes in measures through integrations and transformations.

Positive Measures

Positive measures are a central concept in measure theory, representing a generalized way of assigning 'size' or 'volume' to subsets of a given space. A measure \(u\) is said to be positive if, for every measurable set \(A\), \(u(A) \geq 0\). This non-negativity condition ensures that we are extending the idea of length, area, or volume to more abstract constructions.
Positive measures are introduced as a way to describe distribution or accumulation over spaces. They can be finite or infinite, discrete or continuous, but the central idea remains that they reflect accumulation of some quantity or property in the space considered.

• Used to consistently define integrals, theorems like Fubini's, and properties such as convergence.
• Helpful in that they provide a basic framework for understanding more complex mathematical constructs.