in such way that the Itô isometry still holds. It can then be extended to all B-integrable processes by localization. This method allows the integral to be defined with respect to any Itô process. For a general semimartingale X, the decomposition X = M + A into a local martingale M plus a finite variation process A can be used. Then, the integral can be shown to exist separately with respect to M and A and combined using linearity, H · X = H · M + H · A, to get the integral with respect to X. The standard Lebesgue–Stieltjes integral allows integration to be defined with respect to finite variation processes, so the existence of the Itô integral for semimartingales will follow from any construction for local martingales. For a càdlàg square integrable martingale M, a generalized form of the Itô isometry can be used. First, the Doob–Meyer decomposition theorem is used to show that a decomposition exists, where N is a martingale and is a right-continuous, increasing and predictable process starting at zero. This uniquely defines , which is referred to as the predictable quadratic variation of M. The Itô isometry for square integrable martingales is then which can be proved directly for simple predictable integrands. As with the case above for Brownian motion, a continuous linear extension can be used to uniquely extend to all predictable integrands satisfying . This method can be extended to all local square integrable martingales by localization. Finally, the Doob–Meyer decomposition can be used to decompose any local martingale into the sum of a local square integrable martingale and a finite variation process, allowing the Itô integral to be constructed with respect to any semimartingale. Many other proofs exist which apply similar methods but which avoid the need to use the Doob–Meyer decomposition theorem, such as the use of the quadratic variation [M] in the Itô isometry, the use of the Doléans measure for submartingales, or the use of the Burkholder–Davis–Gundy inequalities instead of the Itô isometry. The latter applies directly to local martingales without having to first deal with the square integrable martingale case. Alternative proofs exist only making use of the fact that X is càdlàg, adapted, and the set {H · Xt: |H| ≤ 1 is simple previsible} is bounded in probability for each time t, which is an alternative definition for X to be a semimartingale. A