Show that (T, ℬ(T), λ) with T=[0, ∞] and λLebesgue measure is σ-finite (i.e., T=⋃i=i^∞ Ti with λ

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Show that (T, ℬ(T), λ) with T=[0, ∞] and λLebesgue measure...

Show that $(T, \mathcal{B}(T), \lambda)$ with $T=[0, \infty]$ and $\lambda$ Lebesgue measure is $\sigma$-finite (i.e., $T=\bigcup_{i=i}^{\infty} T_i$ with $\lambda\left(T_i\right)<\infty$ for each $T_i$.)

Key Concepts

Sigma-Finiteness

Sigma-finiteness is a property of a measure space whereby the entire space can be expressed as a countable union of measurable subsets, each of which has finite measure. This concept is crucial because many theorems in measure theory assume sigma-finiteness in order to extend results that hold for finite measures to more general spaces.

Lebesgue Measure

The Lebesgue measure is a standard way of assigning a volume to subsets of the real numbers and is defined on the Borel sigma algebra. In problems involving Lebesgue measure, it is essential to understand how it assigns finite measure to bounded intervals, which is often used to demonstrate sigma-finiteness by covering unbounded sets with a countable union of such intervals.

Borel Sigma Algebra

The Borel sigma algebra is the collection of all sets that can be formed from open (or equivalently closed) subsets of a given topological space via countable unions, intersections, and complements. It serves as the domain for the Lebesgue measure in many contexts, making it a fundamental concept in the study of measurable spaces.

Countable Decomposition

Countable decomposition involves breaking down a potentially infinite or complicated set into a countable collection of subsets that are easier to handle, each with desirable properties such as having finite measure. This method is often used to show that a measure space is sigma-finite, by constructing an explicit sequence of sets, like bounded intervals, whose combined union covers the entire space.

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