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I am trying to apply a substitution rule to a Lebesgue integral and get very strange results.

Substitution rule (an instantiation of Fremlin, D.H. (2010), Measure Theory, Volume 2, Theorem 263D): Let $\phi: \mathbb{R} \to \mathbb{R}$ be an injective function with derivative $\phi': \mathbb{R} \to \mathbb{R}$. Let $f: \mathbb{R} \to \mathbb{R}$ be measurable and $\mu: \Sigma_{\mathbb{R}} \to \mathbb{R}$ a measure ($\Sigma_{\mathbb{R}}$ is a $\sigma$-algebra on $\mathbb{R}$). Then, $$ \int_{x \in \phi(\mathbb{R})} f(x) d\mu = \int_{x \in \mathbb{R}} |\phi'(x)|*f(\phi(x)) d\mu $$

I instantiate $\mu$ with the dirac measure $\delta(S) := [0 \in S]$, $f$ with the measurable function $f(x) := [x = 0]$ and $\phi$ with the function $\phi(x) := x+1$. Of course, $\phi'(x) = 1$.

Using this instantiation, I get \begin{align*} 1 &= f(0) \\ &= \int_{x \in \mathbb{R}} f(x) d\delta && \text{property of the dirac measure} \\ &= \int_{x \in \phi(\mathbb{R})} f(x) d\delta && \mathbb{R} = \phi(\mathbb{R}) \\ &= \int_{x \in \mathbb{R}} |\phi'(x)|*f(\phi(x)) d\delta && \text{substitution} \\ &= \int_{x \in \mathbb{R}} 1*f(x+1) d\delta \\ &= f(0+1) && \text{property of the dirac measure} \\ &= 0 \end{align*} What did I do wrong?

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What you wrote doesn't match Theorem 263D in my copy of Fremlin. I think you missed an important sentence at the beginning of Section 263.

Throughout this section, as in the rest of the chapter, $\mu$ will denote Lebesgue measure on $\mathbb R^r$.

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