The trading strategy H i is admissible, being S i a ({mathbb {Q}} -local martingale for every ({mathbb {Q}} -localhcal {P}}).
By Theorem 9(i), an admissible control (u_{cdot}^{varepsilon}) is ε-optimal if mathbb{E} int_{0}^{T}f^{varepsilon}(t)u^{varepsilon}_{t},dt= mathbb {E} biggl[ int_{0}^{T-delta}bigl(1+u_{t}^{varepsilon}bigr)u^{varepsilon}_{t},dt+ int_{T-delta}^{T}u_{t}^{varepsilon},dt biggr]leqvarepsilon and thus if mathbb{E} int_{0}^{T}u^{varepsilon}_{t},dtleq varepsilon/2.
Since I is admissible, we have (B in I).
Then for every ϵ > 0, t > 0 and nonzero z ∈ X, there is a positive integer N such that F x j k − ℓ, z ( t ) > 1 − ϵ. for all j, k ≥ N. Since the set A : = { ( j, k ) ∈ N × N : F x j k − ℓ, z ≤ 1 − ϵ }. is contained in S × S, where S = { 1, 2, 3, …, N − 1 } and the ideal I is admissible, A ∈ I. Hence I F - lim x = ℓ.
Since I is admissible and ((x_{j})) is a lacunary (I_{mu} -statistically convergent sequence of order β defI_{mu} -statistically by using the convergent of (mathsequence we see with the lacunary sequence (theta=(h_{i})), the right hand side belofgs torderwhich completes the proof.
Since I is admissible and ((x_{j})) is lacunary (I_{mu} -summable sequence of order β defI_{mu} -summable{M}), usequence coftinuity and using the lacunary sequence (theta=(h_{i})), we can corderde that (w_{I_{mu}}^{beta}(mathscr{M}, theta) subseteq S_{I_{lambda}}^{alpha}(mathscr{M}, theta)).
Since ((x_{j})) is lacunary (I_{lambda} -statistically convergent sequence of order α defI_{lambda} -statisticallysinconvergentmisequenceby using the coftinuity orderathscr{M}), we can say that S_{I_{lambda}}^{alpha}(mathscr{M},theta) subseteq w_{I_{mu}}^{beta}(mathscr{M}, theta!).
Since ((x_{j})) is lacunary (I_{lambda} -statistically convergent sequence of order α defI_{lambda} -statisticallysinconvergentmisequenceby using the coftinuity orderathscr{M}), it follows that the set on the right hand side with the lacunary sequence (theta=(h_{i})) belongs to I and S_{I_{lambda}}^{alpha}(mathscr{M}, theta)subseteq S_{I_{mu}}^{beta}(mathscr{M}, theta).
Theorem 1.7 holds for any admissible f i ∈ L p i, ϕ i ( X, μ ), i = 1, …, m.
We call a Leray endomorphism if (i) all are admissible and (ii) almost all are trivial.
