That is, if A∈I and A⊆⋃i∈IAi where {Ai:i∈I} is a countable collection of sets in I then \[ \mu(A) \le \sum_{i \in I} \m...That is, if A∈I and A⊆⋃i∈IAi where {Ai:i∈I} is a countable collection of sets in I then μ(A)≤∑i∈Iμ(Ai) Finally, μ is clearly σ-finite on I since μ(a,b]<∞ for a,b∈\R with a<b, and \R is a countable, disjoint union of intervals of this form.
Next recall that the distribution of a real-valued random variable X is symmetric about a point a∈\R if the distribution of X−a is the same as the distribution of a−X. ...Next recall that the distribution of a real-valued random variable X is symmetric about a point a∈\R if the distribution of X−a is the same as the distribution of a−X. Note that the interval [q1,q3] roughly gives the middle half of the distribution, so the interquartile range, the length of the interval, is a natural measure of the dispersion of the distribution about the median.
If the point t approaches t0 from the left, the interval does not include the probability mass at t0 until t reaches that value, at which point the amount at or to the left of t incr...If the point t approaches t0 from the left, the interval does not include the probability mass at t0 until t reaches that value, at which point the amount at or to the left of t increases ("jumps") by amount p0; on the other hand, if t approaches t0 from the right, the interval includes the mass p0 all the way to and including t0, but drops immediately as t moves to the left of t0.
