Sentence examples for Borel function from inspiring English sources

L-space properties of such μ are discussed: if μ is Lp-improving and non-negative, and f is a bounded Borel function, then fμ is Lp-improving.

Ifϕis a Borel function locally bounded above onΩ, and ifΩhas a Green's function, then for each x∈Ω,sup{v(x): vsubharmonic on Ω v⩽ϕ}=inf          .We also investigate to what extent these results carry over to plurisubharmonic functions.

For any locally integrable Borel function f on Rd, any Borel subset A of Rd and any j= 1,2,…,d we have∫A∣∣ ∣∣Rjf(x)∣∣ ∣∣dx⩽K∫RdΨ(∣∣ ∣∣f(x)∣∣ ∣∣ dx+|A|⋅L.

A Borel function is defined analogously (see, e.g., Martin 1977).

In particular, if Φ is a real Borel function, then Φ ( B ) is self-adjoint.

Since is the pointwise limit of continuous functions, it is a Borel function.

Let be two sequences of nonnegative Borel functions on.

Since and are the pointwise limits of continuous functions, they are Borel functions.

Let ρ 0, ρ 1, …, ρ m be nonnegative bounded Borel functions on S n − 1.

Assume that (f_{1},ldots,f_{n},f_{n}) are Borel functions all of which are monotone increasing (or monotone decreasing).

Let be a stochastic sequence defined as in Section 1 and a sequence of nondecreasing and nonnegative Borel functions on.