The Kolmogorov Continuity Theorem is a theorem which states that a suitably 'consistent' collection of finite-dimensional distributions will define a stochastic process. The theorem was discovered by Percy John Daniell, the British Mathematician and was credited to the Soviet mathematician Andrey Nikolaevich Kolmogorov, hence the name the Daniell-Kolmogorov Existence Theorem. It is also called as the Daniell-Kolmogorov extension theorem or the Kolmogorov consistency theorem. This theorem provides existence results for nice probability measures on path (function) spaces.

The Daniell-Kolmogorov extension or consistency theorem is one of the first deep theorems of the stochastic processes theory. It provides existence results for nice probability measures on function spaces.

Ε (|| X_{t} - X_{s} ||^{α}) ≤ c |t-s|^^{1+z},
Let α, ε, c >0. If a d-dimensional process (X_{t})_{t ∈ [0,1]} defined on a probability space (Ω, ƒ, P) satisfies for s,t ∈ [0,1], then there exists a modification of the process (X_{t})_{t∈[0,1]} that is a continuous process and whose paths are γ-Hölder for every γ ∈ [0, ε/α).

We make the proof for d=1 and let the reader extend it as an exercise to the case d ≥ 2.
For n ∈ N, we denote
D_{n} = {k/2^{n}, k=0,...,2^{n}}
and
D = U_{n ∈ N} D_{n}.
Let γ ∈ [0, ε/α). From Chebychev’s inequality:
P (max_{1 ≤ k ≤ 2n} |X_{k/2n}} - X_{k-1/2n}| ≥ 2^{-γ n})
= P (U_{1 ≤ k ≤ 2n} |X_{k/2n}} - X_{k-1/2n}| ≥ 2^{-γ n})
≤ Σ_{k=1}^{2n} P |X_{k/2n}} - X_{k-1/2n}| ≥ 2^{-γ n})
≤ Σ_{k=1}^{2n} E|X_{k/2n}} - X_{k-1/2n}| ≥ 2^{-γ n}) / 2^{-γ α n}
≤ c 2^{-n(ε - γ α)}
Therefore, since γ α < ε, we deduce
Σ_{k=n}^{+∞}Σ P (max_{1 ≤ k ≤ 2n} |X_{k/2n}} - X_{k-1/2n}| ≥ 2^{-γ n}) < +∞.
From the Borel-Cantelli lemma, we can thus find a set Ω* ∈ ƒ such that P (Ω^{*}) = 1 and such that for ω ∈ Ω*, there exists N(ω) such that for n ≥ N(ω),
max_{1 ≤ k ≤ 2n} |X_{k/2n}} - X_{k-1/2n}| ≥ 2^{-γ n}.
In particular, there exists an almost surely finite random variable C such that for every n ≥ 0,
max_{1 ≤ k ≤ 2n} |X_{k/2n}} - X_{k-1/2n}| ≥ C 2^{-γ n}.
We now claim that the paths of the restricted process X / _{Ω*} are consequently γ-Hölder on D. Indeed, let s,t ∈ D, t ≠ s. We can find n ≥ 0 such that
1 / 2^{n+1} ≤ |s-t| ≤ 1/2^{n}.
We now pick an increasing and stationary sequence (s_{k})_{k ≥ n} converging toward s,
such that s_{k} ∈ D_{k} and
|s_{k+1} - s_{k}| = 2^{-(k+1)} or 0.
In the same way, we can find an analogue sequence (t_{k})_{k ≥ n} that converges toward t and such that s_{n} and t_{n} are neighbors in D_{n}. We have then:
X_{t} - X_{s} = Σ_{i=n}^{+∞} (X_{si+1} - X_{si}) + (X_{sn} - X_{tn}) + Σ_{i=n}^{+∞}Σ (X_{ti} - X_{ti+1}),
where the above sums are actually finite.
Therefore,
|X_{t} - X_{s}|
≤ C 2^{{-γ n} + 2 Σ_{k=n}^{+∞} C 2^{-γ(k+1)}
≤ 2C Σ_{k=n}^{+∞} 2^{-γk}
≤ 2C / 1-2^{-γ} 2^{-γ n}
Hence the paths of X/Ω* are γ-Hölder on the set D. For ω ∈ Ω*, let t → X'_{t} (ω) be the unique continuous function that agrees with t → X_{t} (ω) on D. For ω ∉ Ω*, we set X'_{t} (ω) = 0. The process (X'_{t})_{t ∈ [0,1]} is the desired modification of (X_{t})_{t ∈ [0,1]}.