Sentence examples for be a function integrable from inspiring English sources

Let (f(x)) be a function integrable with respect to the measure (mu(cdot)).

Let (f cdot)) be a function integrable with respect to the measure (mu(cdot)).

Let (f cdot)) be a function integrable with respect to the measure μ.

Theorem 4.1 Let f(x) be a function integrable in the sense of Lebesgue in [0, 2π] and periodic with period 2π.

Let (f(x)) be a function integrable with respect to the measure μ, where (xin E_{l}), then, for any Borel set (Bsubset E_{l}), we have {lim_{trightarrowinfty}}P t,x,B =mu(B) and P_{x} biggl{ {lim_{Trightarrowinfty}}frac{1}{T} int _{0}^{T}fbigl(x t bigr),dt= int_{E_{l}}f(x mu(dx) biggr} =1.

Let (rho (cdot )) be a function integrable with respect to the measure (pi (cdot )), then for all (xin mathbb{R}^{d}setminus U) mathbb{P} biggl{ lim_{Trightarrow infty }frac{1}{T} int_{0}^{T} rho bigl(X t bigr),dt= int_{mathbb{R}^{d}}rho (x pi (dx) biggr} =1.  .

Furthermore, if (f( cdot )) is a function integrable with respect to the measure μ, then P biggl{ lim_{t to + infty} frac{1}{t} int_{0}^{t} f bigl(Y s) bigr),ds = int_{E^{l}} f(x mu (dx) biggr} = 1.

Let (fin L^{1}(mathbb{R}^{n})) be a function that is integrable on hyperplanes of (mathbb {R}^{n}).

Let be a positive increasing concave nonlinear function on, and let be a nonnegative integrable function on with.

Let (f,g:[a,b]rightarrow[0,1]) be two synchronous functions, and (p:[a,b]rightarrowmathbb{R}) be a nonnegative integrable function.

Definition C. Let be an integrable function, for One defines a linear functional as (2.22).