Express laws and independence through measures

Try

A random variable is a measurable function on a probability space. That definition connects the probability of an output event to the measure of its inverse image. Two independent fair bits give a concrete space with four equally likely outcomes. Add their parity as a third variable and inspect how pairwise independence can coexist with a deterministic relation.

Predict

Predict whether checking all pairs proves mutual independence of three variables. It does not. In the parity example, knowing both input bits determines the third. Predict also whether a zero covariance would prove independence. It is weaker still: covariance checks one product moment, whereas independence compares whole event collections.

Experiment

List the complete parity model.

U	V	W, parity	Probability
0	0	0	1/4
0	1	1	1/4
1	0	1	1/4
1	1	0	1/4

Make a two-way table for each pair and verify its four entries are equal. Then compare the probability of the first row with the product of its three marginal probabilities. This calculation distinguishes the two definitions without relying on terminology alone.

Understand

A probability space is a measure space with total measure one. A real random variable X is measurable into the Borel sets of the real line. Its distribution is the pushforward probability measure, so expectation is an integral with respect either to the underlying space or to that distribution.

Two sigma-algebras are independent when P(A intersection B)=P(A)P(B) for every event A in the first and B in the second. Random variables are independent when their generated sigma-algebras are independent. Their joint law then agrees with the product of marginal laws on every measurable rectangle. Since both are finite measures, the pi-lambda uniqueness theorem extends agreement to the product sigma-algebra. Conversely, equality of these laws immediately gives the rectangle probability formula.

It is often enough to check generating pi systems. Fix an event A in the first generating pi system. The events B satisfying the independence formula form a lambda system: the whole space works; complements use finite probabilities; disjoint unions use countable additivity. The pi-lambda theorem extends the formula to the second generated sigma-algebra. Now fix any event in that full second sigma-algebra and repeat the argument in the first variable. This two-stage extension is necessary; merely mentioning generators does not complete the proof.

For independent integrable X and Y, Tonelli on the product law first gives E abs (XY)=E abs (X)E abs (Y)< infinity. Fubini then proves E(XY)=E(X)E(Y). For several variables, mutual independence requires the corresponding finite intersection formula, not merely all pairwise formulas. The parity table satisfies every pairwise formula but violates the triple formula.

Apply

Use the product-law viewpoint to calculate the distribution of a measurable function of independent variables. First identify the product measure, then take a pushforward under the function. This separates the independence assumption from the transformation. A deterministic function of one variable generally does not produce an independent second variable merely because the output looks different.

Continue

Measure theory clarifies which statements concern laws, underlying events or particular representatives. Next, compare convergence of random variables in several modes and use countable event estimates to turn summable error probabilities into almost-sure conclusions. The probability-one setting supplies finiteness assumptions used earlier in convergence theorems.

Continue to Compare convergence modes using event bounds.