Existence and construction of the solutions of some Markov moment problems are discussed. Starting from the moments of a solution, one recalls a method of recovering this solution, also solving approximately related systems with infinite many nonlinear equations and infinite unknowns. This is the first aim of this review paper. Extension of linear forms with two constraints is applied. Measure theory arguments play a central role. Other results in analysis and functional analysis are used tacitly, sending the reader to the references for unproved stated theorems. Secondly, in the end, existence of solutions of special Markov moment problems is studied.
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