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We present a constructive proof of the fact, that for any subset    ⊆ ℝ  m   and a countable family  ℱ  of bounded functions  f  :    → ℝ there exists a compactification    ′  ⊂  ℓ  2  of    such that every function  f  ∈  ℱ  possesses a continuous extension to a function  f ̅ :   ′→ℝ. However related to some classical theorems, our result is direct and hence applicable in Calculus of Variations. Our construction is then used to represent limits of weakly convergent sequences { f ( u   ν  )}  via  methods related to DiPerna-Majda measures. In particular, as our main application, we generalise the Representation Theorem from the Calculus of Variations due to Kałamajska.

on a certain compactification of an arbitrary subset of ℝm and its applications to DiPerna-Majda measures theory

on a certain compactification of an arbitrary subset of ℝ^m and its applications to DiPerna-Majda measures theory