 In fact, it is always possible to take as the underlying probability space formula_23 and to take for formula_20 the canonical process formula_25.Therefore, an alternative way of stating Kolmogorov's extension theorem is that, provided that the above consistency conditions hold, there exists a (unique) measure formula_26 on formula_27 with marginals formula_5 for any finite collection of times formula_22.Kolmogorov's extension theorem applies when formula_1 is uncountable, but the price to pay
for this level of generality is that the measure formula_26 is only defined on the product σ-algebra of formula_27, which is not very rich.