Question

If $\lambda$ and $\mu$ are positive, mutually singular Borel measures on ${\mathbb{R}}^n$ and $\lambda +\mu$ is regular, then so are $\lambda$ and $\mu$.

Short answer

Both $\lambda$ and $\mu$ are regular because their regularity follows from their mutual singularity and the regularity of $\lambda +\mu$.

Step-by-step solution

Understand the Problem

We are given two positive, mutually singular Borel measures $\lambda$ and $\mu$ on ${\mathbb{R}}^n$, and we know that their sum $\lambda +\mu$ is a regular measure. We need to show that both $\lambda$ and $\mu$ are regular measures as well.

Define Regular Measure

A measure $u$ on a topological space $X$ is called regular if for every Borel set $B$, $u(B)=inf\{u(U):B\subseteq U,U\text{ is open}\}=sup\{u(K):K\subseteq B,K\text{ is compact}\}$. This essentially ensures that the measure can be approximated by measures of open and compact sets.

Use Properties of Mutual Singularity

Since $\lambda$ and $\mu$ are mutually singular, there are disjoint Borel sets $A$ and $B$ such that $\lambda$ is supported on $A$ (i.e., $\lambda (B)=0$) and $\mu$ is supported on $B$ (i.e., $\mu (A)=0$).

Show Regularity of $\lambda$

To show that $\lambda$ is regular, we take any Borel set $C$. Since $\lambda$ is supported on $A$, $\lambda (C)=\lambda (C\cap A)$. For $\lambda +\mu$, we approximate $C\cap A$ by open sets $U$ and compact sets $K$ from within $A$, based on the regularity of $\lambda +\mu$. Since $\mu (K)=\mu (U)=0$ for sets in $A$, the regularity condition for $\lambda (C)$ follows as if we're examining with respect to $A$ alone.

Show Regularity of $\mu$

Similarly, to show that $\mu$ is regular, use the fact that it is supported on $B$. Take any Borel set $D$. Since $\mu$ is supported on $B$, $\mu (D)=\mu (D\cap B)$. Again, approximate $D\cap B$ by open and compact sets within $B$ using the regularity of $\lambda +\mu$. Because $\lambda (K)=\lambda (U)=0$ for sets in $B$, the approximation condition holds as needed for $\mu$ when considering sets in $B$.

Conclude Regularity

The regularity of both $\lambda$ and $\mu$ follows directly from their mutual singularity with respect to the set $A\cup B$ and the regularity of the combined measure $\lambda +\mu$.

Key concepts

Borel Measures

In the realm of mathematics, particularly in measure theory, Borel measures play a critical role. A Borel measure is defined on the Borel $ext\sigma -algebra$ of a topological space. The Borel $ext\sigma -algebra$ is the smallest $ext\sigma -algebra$ that contains all the open sets in a given topological space. It's essentially all sets that can be made from a union, intersection, or complement of open sets.

Borel measures enable us to rigorously assign sizes or 'measures' to these sets, beyond just open sets, extending out to more complex configurations.

When considering measures on ${\mathbb{R}}^n$, a typical real-world application involves understanding the distribution of data points or physical phenomena that can be continuous or scattered over a space. A measure is a way of encapsulating this distribution quantitatively.

One of the fascinating properties of Borel measures is their ability to be regular, which enhances their applications in various mathematical and physical fields. Regularity ensures that the complexity of Borel measures aligns neatly with the nicely-behaved open or compact sets, thus allowing for easier manipulation and approximation.

Mutually Singular Measures

Mutually singular measures arise when we have two measures that do not "see" each other's mass or essentially exist in separate "universes". In more formal terms, if $\lambda$ and $\mu$ are measures on a space, they are called mutually singular if there exist two disjoint sets such that $\lambda$ is concentrated entirely on one set (let's call it $A$) and $\mu$ is concentrated entirely on the other ( ext{$B$}).

This means that for any element in $A$, $\mu$ assigns an area of zero, and vice versa for $B$ and $\lambda$.

• This concept is helpful in distinguishing between different components or parts of measures in analysis.
• It simplifies computations and theoretical considerations by allowing each measure to be treated independently when solving problems related to measure theory or probability.

This type of separation property is crucial when proving regularity of individual measures when their sum is known to be regular, as in the original exercise we are considering.

Topological Space Regularity

When dealing with measures, regularity is a property that provides a bridge between abstract measure theory and concrete calculations. A measure on a topological space is called regular when it allows us to approximate the measure of any set by 'nice' sets: namely, open and compact sets.

More precisely, for any Borel set $B$, the measure is calculated by finding the smallest measure of bigger open sets and the largest measure of smaller compact sets. Mathematically, this means:

$$u(B)=inf\{u(U):B\subseteq U,U\text{ is open}\}=sup\{u(K):K\subseteq B,K\text{ is compact}\}$$

This property is quite powerful because it allows theoretical and computational access to measures by focusing only on open and closed sets.

Regularity simplifies working with Borel measures because it assures us that we can deal with Borel sets using only more manageable and familiar sets like open and compact sets, often leading to clearer and more intuitive results.

Such regular measures are invaluable in analysis, probability, and even in engineering, where precise and predictable behavior of complex systems is critical. Understanding regularity helps to simplify complex mathematical proofs and applications, as highlighted in the original exercise.