Question

Consider (x, p(x)), the discrete sigma-algebra on x. let b∈ x. define µ :p(x)→ r by; µ(e) = . show that µ is measure.

Answer 1

To show that μ is a measure, we need to demonstrate three properties: non-negativity, countable additivity, and null set assignment.

Step-by-step explanation:

Mathematics:

In this question, we are given a random variable x and its corresponding probability values represented by p(x). We need to define a measure, μ, which maps the elements of the sigma-algebra on x to the real numbers. To show that μ is a measure, we need to demonstrate three properties: non-negativity, countable additivity, and null set assignment. Let's go step-by-step:

1. Non-negativity: For any set A in p(x), the measure μ(A) is always greater than or equal to 0.
2. Countable additivity: For any countable sequence of disjoint sets A1, A2, A3,... in p(x), μ(A1 ∪ A2 ∪ A3 ∪ ...) is equal to the sum of the measures of all the individual sets A1, A2, A3,....
3. Null set assignment: The measure μ(∅) of the empty set is equal to 0.

By demonstrating these properties, we can conclude that μ is a measure.