\(N^{ext}_\varepsilon (A) \le N^{int}_\varepsilon (A)\).

The unit interval \(I = [0, 1] \subseteq \mathbb {R}\) has bounded internal covering number with constant \(1\) and exponent \(1\): for \(\varepsilon \le 1\), \(N^{int}_\varepsilon (I) \le 1/\varepsilon \).

Let \(S\) be a finite set of \(E\) and \(x \in E\). Then for all \(y \in S\), \(d_E(x, \pi (x, S)) \le d_E(x, y)\).

Let \(C_\varepsilon \) be a finite \(\varepsilon \)-cover of \(A \subseteq E\) (assuming such a finite cover exists). Then for all \(x \in A\), \(d_E(x, \pi (x, C_\varepsilon )) \le \varepsilon \).

Let \((\varepsilon _n)_{n \in \mathbb {N}}\) be a sequence of positive numbers, \(C_n\) a finite \(\varepsilon _n\)-cover of \(A \subseteq E\) with \(C_n \subseteq A\) and \(x \in C_k\) for some \(k \in \mathbb {N}\). Then for all \(i \le k\), \(\bar{x}_i\in C_i\).

Let \((\varepsilon _n)_{n \in \mathbb {N}}\) be a sequence of positive numbers, \(C_n\) a finite \(\varepsilon _n\)-cover of \(A \subseteq E\) with \(C_n \subseteq A\) and \(x \in C_k\) for some \(k \in \mathbb {N}\). Then for all \(i {\lt} k\), \(d_E(\bar{x}_i, \bar{x}_{i+1}) \le \varepsilon _i\).

Let \((\varepsilon _n)_{n \in \mathbb {N}}\) be a sequence of positive numbers, \(C_n\) a finite \(\varepsilon _n\)-cover of \(A \subseteq E\) with \(C_n \subseteq A\) and \(x \in C_k\) for some \(k \in \mathbb {N}\). Then for \(m \le k\), \(d_E(\bar{x}_m, x) \le \sum _{i=m}^{k-1} \varepsilon _i\).

Let \((\varepsilon _n)_{n \in \mathbb {N}}\) be a sequence of positive numbers, \(C_n\) a finite \(\varepsilon _n\)-cover of \(A \subseteq E\) with \(C_n \subseteq A\). Let \(m, k, \ell \in \mathbb {N}\) with \(m \le k\) and \(m \le \ell \) and let \(x \in C_k\) and \(y \in C_\ell \). Then

\(a^{r_{V,t}-1} \le \vert B_V(t, (r_{V,t}-1)c) \vert \) .

\(\vert B_V(t, r_{V,t}c) \vert \le a^{r_{V,t}}\) .

The radius of a log-size ball sequence \((V_i, t_i, r_i)_{i \in \mathbb {N}}\) for \((J, a, c, n)\) satisfies \(r_i \le n\) for all \(i \in \mathbb {N}\).

The sets \(V_i\) of a log-size ball sequence \((V_i, t_i, r_i)_{i \in \mathbb {N}}\) are a decreasing sequence of sets. That is, \(V_{i+1} \subseteq V_i\) for all \(i \in \mathbb {N}\).

For any log-size ball sequence \((V_i, t_i, r_i)_{i \in \mathbb {N}}\) for \((J, a, c, n)\), for all \(k \ge \vert J \vert \), \(V_k = \emptyset \).

For \(i \ne j\), the balls \(B_{V_i}(t, (r_i-1)c)\) and \(B_{V_j}(t_j, (r_j-1)c)\) of a log-size ball sequence \((V_i, t_i, r_i)_{i \in \mathbb {N}}\) are disjoint.

The cardinal of the pair set \(K\) of a log-size ball sequence for \((J, a, c, n)\) satisfies \(|K| \le a |J|\).

Let \((s, t)\) be a pair in the pair set \(K\) of a log-size ball sequence for \((J, a, c, n)\). Then \(d_T(s, t) \le c n\).

Let \(K\) be the pair set of a log-size ball sequence \((V_i, t_i, r_i)_{i \in \mathbb {N}}\) for \((J, a, c, n)\). Then for any function \(f : T \to E\) with \((E,d_E)\) a metric space,

Let \((T,d_T)\) be a metric space. Let \(J \subseteq T\) be finite, \(a {\gt} 1\), \(c{\gt}0\) and \(n \in \{ 1, 2, ...\} \) such that \(|J| \le a^n\). Then, there is \(K \subseteq J^2\) such that for any function \(f : T \to E\) with \((E,d_E)\) a metric space,

If \(X : T \to \Omega \to E\) is a process that satisfies the Kolmogorov condition, then for all \(t \in T\), \(X_t\) is \(\mathbb {P}\)-a.e. measurable.

If \(E\) is separable and \(X : T \to \Omega \to E\) is a process such that \(X_t\) is \(\mathbb {P}\)-a.e. measurable for all \(t \in T\), then for all \(s, t \in T\), the pair \((X_s, X_t)\) is \(\mathbb {P}\)-a.e. measurable for the Borel \(\sigma \)-algebra on \(E^2\).

If \(X : T \to \Omega \to E\) is a process that satisfies the Kolmogorov condition, then for all \(s,t \in T\) the function \(\omega \mapsto d_E(X_s(\omega ), X_t(\omega ))\) is \(\mathbb {P}\)-a.e. measurable.

Let \(X : T \to \Omega \to E\) be a process that satisfies the Kolmogorov condition for exponents \((p,q)\) with constant \(M\). Let \(\varepsilon {\gt} 0\) and \(C \subseteq T^2\) be a finite set such that for all \((s, t) \in C\), \(d_T(s, t) \le \varepsilon \). Then

Let \(X : T \to \Omega \to E\) be a process that satisfies the Kolmogorov condition for exponents \((p,q)\) with constant \(M\). Let \(J \subseteq T\) be finite, \(a, c \in \mathbb R_+\) with \(a \ge 1\) and \(n \in \{ 1, 2, ...\} \) such that \(|J| \le a^n\). Then

Let \(X : T \to E\). Let \((\varepsilon _n)_{n \in \mathbb {N}}\) be a sequence of positive numbers, \(C_n\) a finite \(\varepsilon _n\)-cover of \(J \subseteq T\) with \(C_n \subseteq J\). For \(m \le k\),

Let \(X : T \to \Omega \to E\) be a process that satisfies the Kolmogorov condition for exponents \((p,q)\) with constant \(M\). Let \(C\) be a finite \(\varepsilon \)-cover of \(J \subseteq T\) with \(C \subseteq J\), with minimal cardinal. Then for \(c \ge 0\),

Note the logarithm has base \(2\).

Let \(X : T \to \Omega \to E\) be a process that satisfies the Kolmogorov condition for exponents \((p,q)\) with constant \(M\). For all \(n \in \mathbb {N}\), let \(C_n\) a finite \(\varepsilon _n\)-cover of \(J \subseteq T\) with \(C_n \subseteq J\) for \(\varepsilon _n = \varepsilon _0 2^{-n}\), with minimal cardinal. Let \(\delta {\gt} 0\) and let \(n_2\) be such that \(\varepsilon _0 2^{-n_2} {\lt} \delta /2 \le \varepsilon _0 2^{-n_2+1}\). Let \(k = \min \{ j \in \mathbb {Z} \mid \varepsilon _0 2^{-j} {\lt} \inf _{s, t \in J; s \ne t}d_T(s, t)\} \). For \(m = \min (n_2, k)\),

TODO: \(C_k = J\)

Let \(X : T \to \Omega \to E\) be a process that satisfies the Kolmogorov condition for exponents \((p,q)\) with constant \(M\). Let \((\varepsilon _n)_{n \in \mathbb {N}}\) be a sequence of positive numbers and \(C_n\) a finite \(\varepsilon _n\)-cover of \(T\) with \(C_n \subseteq T\). Then for \(j {\lt} k\),

Let \(X : T \to \Omega \to E\) be a stochastic process. Let \((\varepsilon _n)_{n \in \mathbb {N}}\) be a sequence of positive numbers and \(C_n\) a finite \(\varepsilon _n\)-cover of \(T\) with \(C_n \subseteq T\). For \(m \le k\),

Let \(X : T \to \Omega \to E\) be a process that satisfies the Kolmogorov condition for exponents \((p,q)\) with constant \(M\). Let \((\varepsilon _n)_{n \in \mathbb {N}}\) be a sequence of positive numbers and \(C_n\) a finite \(\varepsilon _n\)-cover of \(T\) with \(C_n \subseteq T\). Then for \(m \le k\),

Let \(X : T \to \Omega \to E\) be a process that satisfies the Kolmogorov condition for exponents \((p,q)\) with constant \(M\). Let \((\varepsilon _n)_{n \in \mathbb {N}}\) be a sequence of positive numbers in \((0, \mathrm{diam}(T))\) and \(C_n\) a finite \(\varepsilon _n\)-cover of \(T\) with \(C_n \subseteq T\), and with minimal cardinality. Suppose that \(T\) has bounded internal covering number with constant \(c_1{\gt}0\) and exponent \(d {\gt} 0\). Then for \(m \le k\),

Let \(J \subseteq T\) be a finite set and suppose that \(T\) has finite diameter. For \(k \in \mathbb {N}\), let \(\eta _k = 2^{-k}(\mathrm{diam}(T) + 1)\). For \(X : T \to \Omega \to E\) a stochastic process and \(\beta \in (0, (q - d)/p)\),

Suppose that \(J \subseteq T\) is a finite set and that \(T\) has bounded internal covering number with constant \(c_1{\gt}0\) and exponent \(d {\gt} 0\). Let \(X : T \to \Omega \to E\) be a process that satisfies the Kolmogorov condition for exponents \((p,q)\) with constant \(M\), with \(q {\gt} d\) and \(p {\gt} 0\). Let \(\beta \in (0, (q - d)/p)\). Then there exists a constant \(L(T, c_1, d, p, q, \beta ) {\lt} \infty \) (that depends on \(T\) bot not \(J\)) such that

Under the assumptions of Theorem 4.48, for \(E\) a complete space and \(\beta \in (0, (q - d)/p)\), there exists a version \(Y\) of \(X\) (i.e., a process \(Y\) with \(\mathbb {P}(Y_t \ne X_t) = 0\) for all \(t\)) such that the paths of \(Y\) are Hölder continuous of order \(\beta \).

\(\mathbb {R}_+\) has a cover with bounded covering numbers for the sets \(T_n = [0,n)\), constants \(c_n = n\) and exponents \(d_n = 1\).

Let \(T\) be a metric space with a cover \((T_n)\) with bounded covering numbers with constants \(c_n\) and exponents \(d_n\). Let \(X : T \to \Omega \to E\) be a process that satisfies the Kolmogorov condition for exponents \((p,q)\) with constant \(M\), with \(q {\gt} \sup _n d_n\). Then \(X\) satisfies the localized Kolmogorov conditions with exponents \((p_n, q_n) = (p, q)\), constants \(M_n = M\) and any \(\rho _n {\gt} 0\).