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# 数学代写|实分析代写Real Analysis代考|MATH331

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## 数学代写|实分析代写Real Analysis代考|Change of Variables for the Lebesgue Integral

A general-looking change-of-variables formula for the Riemann integral was proved in Section III.10. On closer examination of the theorem, we found that the result did not fully handle even as ostensibly simple a case as the change from Cartesian coordinates in $\mathbb{R}^2$ to polar coordinates. Lebesgue integration gives us methods that deal with all the unpleasantness that was concealed by the earlier formula.

Theorem 6.32 (change-of-variables formula). Let $\varphi$ be a one-one function of class $C^1$ from an open subset $U$ of $\mathbb{R}^N$ onto an open subset $\varphi(U)$ of $\mathbb{R}^N$ such that $\operatorname{det} \varphi^{\prime}(x)$ is nowhere 0 . Then
$$\int_{\varphi(U)} f(y) d y=\int_U f(\varphi(x))\left|\operatorname{det} \varphi^{\prime}(x)\right| d x$$
for every nonnegative Borel function $f$ defined on $\varphi(U)$.
REMARK. The $\sigma$-algebra on $\varphi(U)$ is understood to be $\mathcal{B}_N \cap \varphi(U)$, the set of intersections of Borel sets in $\mathbb{R}^N$ with the open set $\varphi(U)$. If $f$ is extended from $\varphi(U)$ to $\mathbb{R}^N$ by defining it to be 0 off $\varphi(U)$, then measurability of $f$ with respect to this $\sigma$-algebra is the same as measurability of the extended function with respect to $\mathcal{B}_N$.

Proof. Theorem 3.34 gives us the change-of-variables formula, as an equality of Riemann integrals, for every $f$ in $C_{\text {com }}(\varphi(U))$. In this case the integrands on both sides, when extended to be 0 outside the regions of integration, are continuous on all of $\mathbb{R}^N$, and the integrations can be viewed as involving continuous functions on compact geometric rectangles. Proposition 6.11 (or Theorem 6.31 if one prefers) allows us to reinterpret the equality as an equality of Lebesgue integrals.
In the extension of this identity to all nonnegative Borel functions, measurability will not be an issue. The function $f$ is to be measurable with respect to $\mathcal{B}_N(\varphi(U))$, and Corollary 6.29 shows that such $f$ ‘s correspond exactly to functions $f \circ \varphi$ measurable with respect to $\mathcal{B}_N(U)$.
Using Theorem 5.19 , define a measure $\mu$ on $\mathcal{B}_N(U)$ by
$$\mu(E)=\int_E\left|\operatorname{det} \varphi^{\prime}(x)\right| d x$$

## 数学代写|实分析代写Real Analysis代考|Hardy-Littlewood Maximal Theorem

This section takes a first look at the theory of almost-everywhere convergence. The theory developed historically out of Lebesgue’s work on an extension of the Fundamental Theorem of Calculus to general integrable functions on intervals of the line, work that we address largely in the next chapter. We shall see gradually that the theory applies to a broader range of problems than the ones immediately generalizing Lebesgue’s work, and one can make a case that nowadays the theory in this section is of considerably greater significance in real analysis than one might expect from Lebesgue’s work on the Fundamental Theorem.

The theory brings together two threads. The first thread is the observation that an effort to differentiate integrals of general integrable functions on an interval of the line can be reinterpreted as a problem of almost-everywhere convergence in connection with an approximate identity of the kind in Theorem 6.20. In explaining this assertion, let us denote Lebesgue measure by $m$ as necessary. To differentiate $F(x)=\int_a^x f(t) d t$, one forms the usual difference quotient $h^{-1}[F(x+h)-F(x)]$, which can be written for $h>0$ as
$$\frac{1}{m([-h, 0])} \int_{[-h, 0]} f(x-y) d y=\int_{\mathbb{R}^1} f(x-y) m([-h, 0])^{-1} I_{[-h, 0]}(y) d y$$
or as $f * \varphi_h(x)$, where $\varphi(y)=m([-1,0])^{-1} I_{[-1,0]}(y)$. Here $\varphi$ has integral 1 , and $\varphi_h$ is the normalized dilated function defined in Section 2 by $\varphi_h(y)=h^{-1} \varphi\left(h^{-1} y\right)$ in the 1-dimensional case. Theorem 6.20 says for $p=1$ and $p=2$ that as $h$ decreases to $0, f * \varphi_h$ converges to $f$ in $L^p$ if $f$ is in $L^p$. Also, $f * \varphi_h$ converges uniformly to $f$ if $f$ is bounded and uniformly continuous, and $f * \varphi_h(x)$ converges to $f(x)$ at the point $x$ if $f$ is bounded and is continuous at $x$. The problem about differentiation of integrals asks about convergence almost everywhere.

We shall want to have a theorem in $\mathbb{R}^N$, and for this purpose an $N$-dimensional version of $I_{[-1,0]}$ does not seem attractive for generalizing. Instead, let us generalize from $I_{[-1,1]}$, taking the $N$-dimensional problem to involve a ball $B$ of radius 1 centered at the origin; there is some flexibility in choosing the set $B$, and a cube centered at the origin would work as well. We write $r B$ for the set of dilates of the members of $B$ by the scalar $r$. Thus we investigate
$$m(r B)^{-1} \int_{r B} f(x-y) d y$$
as $r$ decreases to 0 ; equivalently we investigate
$$f * \varphi_r(x), \quad \text { where } \varphi(y)=m(B)^{-1} I_B(y) .$$

## 数学代写|实分析代写Real Analysis代考|Change of Variables for the Lebesgue Integral

$$\int_{\varphi(U)} f(y) d y=\int_U f(\varphi(x))\left|\operatorname{det} \varphi^{\prime}(x)\right| d x$$

$$\mu(E)=\int_E\left|\operatorname{det} \varphi^{\prime}(x)\right| d x$$

## 数学代写|实分析代写Real Analysis代考|Hardy-Littlewood Maximal Theorem

$$\frac{1}{m([-h, 0])} \int_{[-h, 0]} f(x-y) d y=\int_{\mathbb{R}^1} f(x-y) m([-h, 0])^{-1} I_{[-h, 0]}(y) d y$$

$$m(r B)^{-1} \int_{r B} f(x-y) d y$$

$$f * \varphi_r(x), \quad \text { where } \varphi(y)=m(B)^{-1} I_B(y) .$$

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