This paper introduces notions of multiresolution analysis (MRA) and frame multiresolution analysis (FMRA) in a reducing subspace \(X_\Omega\) of \(L^2(\mathbb{R})\) (see Definition~2), where \(X_\Omega:=\{ f\in L^2(\mathbb{R}) : \text{Supp}(\hat f)\subset \Omega\}\) and \(\Omega\) is a measurable subset of \(\mathbb{R}\) such that \(\Omega=2\Omega\). One particular interesting example of the reducing space \(X_\Omega\) is \(\Omega=[0,\infty)\). Refinable functions and frame wavelets in a reducing space \(X_\Omega\) are discussed in the paper. Many examples of refinable functions and frame wavelets derived from MRA's and FMRA's are presented to illustrate the general theory. The reviewer points out that complete characterizations for tight wavelet frames, dual wavelet frames, and (bi)orthogonal wavelets in a general reducing subspace \(X_\Omega\) in any dimension have been established by \textit{B. Han} [Appl. Comput. Harmon. Anal. 4, No. 4, 380--413 (1997; Zbl 0880.42017)].