invariant subalgebra sentence in Hindi

Examples

1. Measurable sets invariant mod 0 under a flow or a semigroup action form the "'invariant subalgebra "'of ?, and the corresponding measure-preserving dynamical system is ergodic if the invariant subalgebra is the trivial " ? "-algebra consisting of the sets of measure 0 and their complements in " X ".
2. Measurable sets invariant mod 0 under a flow or a semigroup action form the "'invariant subalgebra "'of ?, and the corresponding measure-preserving dynamical system is ergodic if the invariant subalgebra is the trivial " ? "-algebra consisting of the sets of measure 0 and their complements in " X ".
3. The dual Hopf algebra T introduced above as well in the Hopf algebroid context and the dual left action becomes a right coaction that makes A a T-Galois extension of B . The condition that the Frobenius homomorphism map A onto all of B is used to show that B is precisely the invariant subalgebra of the Hopf-Galois action ( and not just contained within ).