answer:Given the information in the original scenario, it is stated that a strong GPA is necessary for law school acceptance, and if Tom does well this semester, he will have a strong enough GPA to be accepted to law school. In this case, we can conclude that doing well this semester and getting a strong enough GPA is indeed sufficient for Tom to be accepted to law school, based on the information provided. The phrase strong enough GPA to be accepted to law school implies that meeting this condition will meet the GPA requirement for acceptance. While being part of the debate team is highly valued by law schools, the scenario does not indicate that it is a necessary condition for acceptance. Therefore, we can tell Tom that if he does well this semester and achieves a strong enough GPA, it will be sufficient to guarantee his acceptance to law school, based on the information provided in the scenario.

answer:The Lebesgue Outer Measure is a measure defined for any subset of the real numbers and is used to extend the concept of length for intervals to more general sets. It is a non-negative extended real number (including infinity) that provides an upper bound for the 'size' of a set. The Lebesgue Outer Measure of a set A, denoted by μ*(A), is defined as the infimum (smallest value) of the sums of lengths of intervals that cover A. A covering of a set A is a collection of intervals whose union contains A. The lengths of the intervals are added together, and the smallest possible sum obtained from the various coverings is the outer measure. Formally, it can be expressed as follows: μ*(A) = inf{ ∑ |In| : {In} is a covering of A by intervals}. Here, 'inf' represents the infimum, and the summation is over all intervals In in the covering. The Lebesgue Outer Measure provides a way to measure sets in a more general and flexible manner than traditional notions of length or volume, especially for sets that are not intervals or do not have a clear geometric shape. However, the Lebesgue Outer Measure does not satisfy the properties of a sigma-algebra measure (it is not countably additive for all sets), but it provides the foundation for the definition of the Lebesgue Measure, which does have these desirable properties for Lebesgue measurable sets.

answer:Yes, the Lebesgue Outer Measure satisfies countable subadditivity. This means that for any sequence {Aj} of subsets of the real numbers, the Lebesgue Outer Measure of the union of the sets is less than or equal to the sum of the Lebesgue Outer Measures of the individual sets. Mathematically, this property can be expressed as follows: μ*(∪ Aj) ≤ ∑ μ*(Aj) This property indicates that the Lebesgue Outer Measure behaves reasonably well when dealing with countable unions of sets. However, it is worth noting that the Lebesgue Outer Measure does not satisfy countable additivity (equality instead of inequality) for all sets, which is a more desirable property in a measure. Countable additivity holds for Lebesgue measurable sets, which is a more restricted class of sets. Countable subadditivity is a key property that makes the Lebesgue Outer Measure useful for defining the Lebesgue Measure, which is a countably additive measure restricted to Lebesgue measurable sets.

answer:Caratheodory's Criterion for Lebesgue measurability states that a set A is Lebesgue measurable if and only if for every set E, the following equation holds: μ*(E) = μ*(E ∩ A) + μ*(E ∩ A^c) where A^c is the complement of the set A. This criterion uses the Lebesgue Outer Measure μ* to characterize Lebesgue measurable sets. It is a necessary and sufficient condition for a set to be Lebesgue measurable. The idea behind Caratheodory's Criterion is that a set A is Lebesgue measurable if it splits every set E into two parts, E ∩ A and E ∩ A^c, in such a way that the Lebesgue Outer Measure of E is equal to the sum of the Lebesgue Outer Measures of these two parts. This criterion provides a useful way to determine whether a set is Lebesgue measurable, and it is often used in measure theory to prove the measurability of certain sets.