Question

Let \((X, \mathcal{M}, \mu)\) be a measure space. A set \(E \in \mathcal{M}\) is called locally null if \(\mu(E \cap F)=0\) for every \(F \in \mathcal{M}\) such that \(\mu(F)<\infty\). If \(f: X \rightarrow \mathbb{C}\) is a measurable function, define $$ \|f\|_{*}=\inf \\{a:\\{x:|f(x)|>a\\} \text { is locally null }\\}, $$ and let \(\mathcal{L}^{\infty}=\mathcal{L}^{\infty}(X, \mathcal{M}, \mu)\) be the space of all measurable \(f\) such that \(\|f\|_{0}<\infty\). We consider \(f, g \in \mathcal{L}^{\infty}\) to be identical if \(\\{x: f(x) \neq g(x)\\}\) is locally null. a. If \(E\) is locally null, then \(\mu(E)\) is cither 0 or \(\infty\). If \(\mu\) is semifinite, then every locally null set is null. b. \(\|\cdot\|_{*}\) is a norm on \(\mathcal{L}^{\infty}\) that makes \(\mathcal{L}^{\infty}\) into a Banach space. If \(\mu\) is semifinite, then \(\mathcal{L}^{\infty}=L^{\infty}\).

Short answer

a. Locally null sets are either null or infinite. b. \(\|\cdot\|_{*}\) defines a norm making \(\mathcal{L}^{\infty}\) a Banach space; it equals \(L^{\infty}\) if \(\mu\) is semifinite.

Step-by-step solution

Understanding Locally Null Sets and Measures

A set \(E \in \mathcal{M}\) is considered locally null if the measure \(\mu(E \cap F) = 0\) for any set \(F \in \mathcal{M}\) where \(\mu(F) \lt \infty\). This means that \(E\) \'vanishes\' at every point where \(F\) has finite measure. Under these conditions, if \(\mu(E)\) is not 0, it must be infinite, which contrasts with local nullity by being entirely unrestricted when intersecting with finite measure sets.

Understanding Semi-finite Measures

A measure \(\mu\) is semifinite if, for every set \(E \in \mathcal{M}\) with \(\mu(E) = \infty\), there exists a subset \(F \subseteq E\) such that \(\mu(F) \lt \infty\) and \(F\in\mathcal{M}\). Therefore, under a semifinite measure, any locally null set \(E\) must actually be a null set because it cannot have a nonzero measure without contradiction.

Defining the Norm \(\|\cdot\|_{*}\)

For a measurable function \(f: X \to \mathbb{C}\), the norm \(\|f\|_{*}\) is defined as the infimum of values \(a\) such that the set \(\{x : |f(x)| > a\}\) is locally null. This measures the \'smallest\' maximum value of \(|f(x)|\) for which the set where \(f\) exceeds this value is insignificant in terms of measure, hence locally null.

Proving the Space \(\mathcal{L}^{\infty}\) is a Banach Space

To prove \(\|\cdot\|_{*}\) is a norm, we must show it satisfies positivity, scalability, and the triangle inequality. Positivity and scalability follow from the definition, asserting zero only when \(f = 0\) almost everywhere, and \(\|cf\|_{*} = |c|\|f\|_{*}\). For the triangle inequality, given \(f,g\in\mathcal{L}^{\infty}\), \(\|f + g\|_{*} \leq \|f\|_{*} + \|g\|_{*}\) since the set where \((f+g)\) is comparable is covered by \(\{x:|f(x)|>a\}\) and \(\{x:|g(x)|>b\}\). Then each locally null, and hence the union is locally null. Completeness needs verifying, for which convergence implies uniform boundedness within \(L^{\infty}\) due to the finite supremum chosen for \(\|f_n-f_m\|_*\) converging to 0 in the nature of its norm.

Relation Between \(\mathcal{L}^{\infty}\) and \(L^{\infty}\) Under Semifinate Measure

When \(\mu\) is semifinite, every locally null set is null. Therefore, the distinction between \(\mathcal{L}^{\infty}\) and \(L^{\infty}\) vanishes, because the exceptional sets claimed as locally null under \(\|f\|_*\) norm consideration translate exactly to null sets, meaning \(\mathcal{L}^{\infty} = L^{\infty}\).

Key concepts

Locally Null Sets

A locally null set is an intriguing concept in measure theory. It's a set where, when combined—via intersection—with any set of finite measure, the resulting set has zero measure. Imagine that you are focusing only on where the measure doesn't blow up to infinity. If you have a set \(E\), and you take a part of it that overlaps with any finite measure set \(F\), that overlap should be null, meaning it's essentially tiny enough to be "invisible" under the measure \(\mu\).

Why does this concept matter? Locally null sets help in understanding when we can consider some set as negligible, or "almost non-existent," when intersected with finite measure contexts. If the measure \(\mu(E)\) is not zero, it must jump to infinity because a zero or finite measure can't fit a set that gives finite results in all finite intersections.

Understanding locally null sets provides a strong foundation for more complex topics like the integration of functions over measure spaces. If an exceptional set is locally null, you can often ignore it in mathematical deductions involving finite measures.

Semi-finite Measures

Semi-finite measures offer a framework to ensure that every set of infinite measure has pieces (subsets) that are manageable, i.e., finite. When a measure \(\mu\) is semi-finite, it means that you can always find a smaller chunk of any infinite set that still feels measurable and finite.

Consider you have a set \(E\) with \(\mu(E) = \infty\). Semi-finiteness tells us there exists a subset \(F\) of \(E\) such that \(\mu(F) < \infty\). Now, for a locally null set within this context, it cannot remain infinite without causing a contradiction. Ergo, locally null sets have to also be null, simplifying many aspects of measure theory.

This characteristic is crucial when translating between different spaces, like \(\mathcal{L}^{\infty}\) and \(L^{\infty}\), as it shows under semi-finite measures, these spaces could essentially become the same since their treatment of exceptional sets aligns perfectly.

Banach Space

A Banach space is a complete normed vector space; this means every Cauchy sequence in this space has a limit that also belongs to the space. When discussing \(\mathcal{L}^{\infty}\), which consists of bounded measurable functions, establishing it as a Banach space involves the norm defined by \(\| f \|_{*} \). This norm essentially evaluates the 'smallest' upper bound of function values beyond which lies an insignificant, locally null set.

Here are some key characteristics that make \(\| f \|_{*} \) a norm and \(\mathcal{L}^{\infty}\) a Banach space:

• Positivity: The norm is zero only if the function \(f\) is zero everywhere except possibly on a locally null set.
• Scalability: If you multiply the function by a constant \(c\), the norm scales by \(|c|\).
• Triangle Inequality: The combined value bounds of two functions intersect within locally null sets, preserving compactness.

Combining this with semi-finite measures, the space aligns perfectly with \(L^{\infty}\), a familiar setting in functional analysis, offering powerful tools for analysis and predictions in continuous mathematical models.