Valid generalisation from approximate interpolation

Let ℋ and ℓ be sets of functions from domain X to ℝ. We say that ℋ validly generalises ℓ from approximate interpolation if and only if for each η > 0 and ε δ ∈ (0, 1) there is m₀(η, ε δ) such that for any function t ∈ ℓ and any probability distribution ℘ on X, if m ≥ m₀ then with ℘^m-probability at least 1 - δ, a sample X = (x₁, x₂,...,x_m) ∈ X^m satisfies ∀h ∈ ℋ ιh(x_i) - t(x_i)ι <η, (1 ≤ i ≤m) ⇒ ℘({x :ιh(x) - t(x)ι ≥ η}) < e. We find conditions that are necessary and sufficient for ℋ to validly generalise ℓ from approximate interpolation, and we obtain bounds on the sample length m₀(η ε δ) in terms of various parameters describing the expressive power of ℋ.