Intereting Posts

Homomorphisms and exact sequences
P(A|C)=P(A|B)*P(B|C)?
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Prove $\;\big((p\rightarrow q) \lor (p \rightarrow r)\big) \rightarrow (q\lor r)\equiv p \lor q \lor r$ without use of a truth table.
Little Bézout theorem for smooth functions
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Polynomials representing primes
Is there any matrix notation the summation of a vector?
Improper integral of $\frac{x}{e^{x}+1}$
If $A=AB-BA$, is $A$ nilpotent?
Can I bring the variable of integration inside the integral?
Identify the plane defined by $|z-2i| = 2|z+3|$
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Show that a proper continuous map from $X$ to locally compact $Y$ is closed

This one comes from Folland, Real Analysis, Problem 33 in the section titled Modes of Convergence.

Suppose $f_n \geq 0$ and $f_n \rightarrow f$ in measure, then $\int f \leq \liminf \int f_n$.

So I notice a few things first off, that since $f_n \to f$ in measure, we can find a subsequence $f_{n_j}$ which converges pointwise almost everywhere (Theorem 2.30 in Folland), and for this subsequence we may say (by Fatou’s lemma using $f_n \geq 0$) that $\int f \leq \liminf \int f_{n_j}$, but it’s not necessarily true that $\liminf \int f_{n_j} \leq \liminf \int f_n$, or at least I don’t see how to prove it (and in general this is not true for any sequence and subsequence, while the reverse inequality is, I think).

- Axiom of choice, non-measurable sets, countable unions
- C* algebra of bounded Borel functions
- Translating an integrable function creates a sequence that converges to $0$ almost everywhere
- Can anyone give an example of a closed set contains no interval but with finite non-zero Lebesgue measure?
- Integral of Schwartz function over probability measure
- Assume that $ f ∈ L()$ and $\int x^nf(x)dx=0$ for $n=0,1,2…$.

Any tips, hints, or solutions?

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- Stone's Theorem Integral: Basic Integral
- a.s. Convergence and Convergence in Probability
- Axiom of choice, non-measurable sets, countable unions
- Discontinuity points of a Distribution function
- Proving Caratheodory measurability if and only if the measure of a set summed with the measure of its complement is the measure of the whole space.
- Homogeneous function in bounded mean oscillation BMO($\mathbb R^n$) space
- Show that if $E\subset\mathbb{R}$ is a measurable set, so $f:E\rightarrow \mathbb{R}$ is a measurable function.
- Is Steinhaus theorem ever used in topological groups?

You can pass to a subsequence $f_{n_k}$ with $\int f_{n_k} \to \liminf \int f_n$ first.

This subsequence will also converge to $f$ in measure and … then you already know what to do.

You can use the Urysohn subsequence principle, but it should be modified a little bit.

(Urysohn subsequence principle).Let $x_n$ be a sequence of real numbers, and let $x$ be another real number, then $\liminf x_n\geq x$ iff every subsequence $x_{n_j}$ of $x_n$ has a further subsequence $x_{n_{j_i}}$ such that $\liminf x_{n_{j_i}}\geq x$.

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- Sum $\frac{1}{6} + \frac{5}{6\cdot 12} + \frac{5\cdot 8}{6\cdot 12\cdot 18} + \frac{5\cdot 8\cdot 11}{6\cdot 12\cdot 18\cdot 24}+\ldots$
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