Yu, X., Riesz representation theorem, Borel measures and subsystems of second-order arithmetic, Annals of Pure and Applied Logic 59 65-78. Formalized concept of finite Borel measures is developed in the language of second-order arithmetic. Formalization of the Riesz representation theorem is proved to be equivalent to arithmetical comprehension. Codes of Borel sets of complete separable metric spaces are defined and proved to be meaningful in the subsystem ATR0. Arithmetical transfinite recursion is enough to prove the measurability of Borel sets for (...) any finite Borel measure on a compact complete separable metric space. (shrink)
We develop measure theory in the context of subsystems of second order arithmetic with restricted induction. We introduce a combinatorial principleWWKL (weak-weak König's lemma) and prove that it is strictly weaker thanWKL (weak König's lemma). We show thatWWKL is equivalent to a formal version of the statement that Lebesgue measure is countably additive on open sets. We also show thatWWKL is equivalent to a formal version of the statement that any Borel measure on a compact metric space is countably additive (...) on open sets. (shrink)
Concepts of L1 space, integrable functions and integrals are formalized in weak subsystems of second order arithmetic. They are discussed especially in relation with the combinatorial principle WWKL (weak-weak König's lemma and arithmetical comprehension. Lebesgue dominated convergence theorem is proved to be equivalent to arithmetical comprehension. A weak version of Lebesgue monotone convergence theorem is proved to be equivalent to weak-weak König's lemma.
We study the formalization within sybsystems of second-order arithmetic of theorems concerning periodic points in dynamical systems on the real line. We show that Sharkovsky's theorem is provable in WKL0. We show that, with an additional assumption, Sharkovsky's theorem is provable in RCA0. We show that the existence for all n of n-fold iterates of continuous mappings of the closed unit interval into itself is equivalent to the disjunction of Σ02 induction and weak König's lemma.
Arithmetical comprehension is proved to be equivalent to the enumerability of singular points of any measure on the Cantor space. It is provable in ACA0 that any perfect closed subset of [0, 1] is the support of some continuous positive linear functional on C[0, 1].
Editor's Note: Over a long period of time, the evil bourgeois reactionary line has created antagonism between two groups of students in schools—antagonism on the basis of one's family background. This antagonism became very obvious during the initial stage of the Cultural Revolution, and has lasted to this day. It has prevented further criticism of the bourgeois reactionary line and hindered further development of the Cultural Revolution.
The Rising Sun Battlefield Journal published by the East-Is-Red Commune of the Beijing Institute of Light Industry devoted as many as six pages to an all-out attack of "On Family Background" in a long and despicable article titled "The Big Poisonous Weed ‘On Family Background’ Must Be Torn Up by the Roots." In their own words, the appearance of this article was inevitable at a time when the Cultural Revolution reached the stage of great alliance [of mass rebel organizations] and (...) the great power seizure [from capitalist-roaders within the Party]. (shrink)
In December of last year, a few clowns appeared on the grand and spectacular stage of the Great Proletarian Cultural Revolution. These clowns were the reincarnated ghosts from the Capital Red Guard West City, East City, and Haidian Districts Pickets. They viciously attacked Chairman Mao's revolutionary line, engaged in slander on the Central Cultural Revolution Group, called dear Comrade Jiang Qing names, and sabotaged the organizations under the proletarian dictatorship. They provoked violence, created chaos, searched and confiscated the possessions of (...) revolutionary organizations, and spread reactionary declarations. The crimes they committed are simply too numerous to mention. These clowns were none other than the "Capital Red Guards United Action Committee.". (shrink)