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text:
# $\mu$ measurable functions and separable metric spaces

I was reading a math textbook and the author gives the following without proof. I have no clue on how to proceed.

Let $(X, \mathcal{F}, \mu)$ be a measure space and $(Y,d)$ be a separable metric space ($d$ is the metric). If $f:(X,\mathcal{F}) \rightarrow (Y, d)$ is a $\mu$-measurable function prove that there exists an $\mathcal{F}$ measurable function which coincides with $f$ everywhere except on a $\mu$-negligible set.

Any help is greatly appreciated.

EDIT: The textbook is “Functions of Bounded Variation and Free Discontinuity Problems” by Luigi Ambrosio et. al.

#### Solutions Collecting From Web of "$\mu$ measurable functions and separable metric spaces"

Edit: I have just figured a much easier way.

Let $\mathcal{V} = \{V_n : n = 1, 2, \dotsc\}$
be a countable base for the topology of $Y$.

For each $V_n$, choose a negligible $E_n \subset X$ such that
$f^{-1}(V_n) \setminus E_n \in \mathcal{F}$.
It may happen that $\bigcup E_n \not \in \mathcal{F}$.
But since it is a negligible set, there is a negligible
$Z \in \mathcal{F}$ such that $\bigcup E_n \subset Z$.

Fix some $y \in Y$,
and then define
$$g(x) = \left\{ \begin{array}{} f(x), & x \not \in Z \\ y, & x \in Z \end{array} \right.$$

Notice that for any $V_n \in \mathcal{V}$,
if $y \not \in V_n$,
\begin{align*} g^{-1}(V_n) &= f^{-1}(V_n) \setminus Z \\&= (f^{-1}(V_n) \setminus E_n) \setminus Z \in \mathcal{F}. \end{align*}
And if $y \in V_n$,
\begin{align*} g^{-1}(V_n) &= f^{-1}(V_n) \cup Z \\&= (f^{-1}(V_n) \setminus E_n) \cup Z \in \mathcal{F}. \end{align*}
That is, $g^{-1}(\mathcal{V}) \subset \mathcal{F}$.
All open sets of $Y$ are (countable) union of elements in $\mathcal{V}$.
Therefore, $\mathcal{V}$ generates the $\sigma$-algebra of Borel sets
$\mathcal{B}$.
And so, $g$ is $\mathcal{F}$-measurable.
In fact,
$$g^{-1}(\mathcal{B}) = g^{-1}(\sigma(\mathcal{V})) = \sigma \left(g^{-1}(\mathcal{V})\right) \subset \mathcal{F}.$$

Since it is evident that $g$ and $f$ are equal almost everywhere,
the proof is complete.

Let $(V_n) \in Y$ be a countable base for the separable metric space $Y$. Then define
$V_n^\prime = V_n \cap V_{n-1}^c \cap V_{n-2}^c… \cap V_1^c$ and $V_1^\prime = V_1$. Clearly, $V_n^\prime$ are all
measurable. Now $f^{-1}(V_n^\prime) = E_n \cup N_n = E_n + (E_n^c \cap N_n) = E_n + N_n^\prime$, where $E_n$ is
$\mathcal{F}$-measurable and $N_n, N_n^\prime$ are
$\mu$-negligible. As all $V_n^\prime$ are all pairwise disjoint, $f^{-1}(V_n^\prime) = E_n + N_n^\prime$ are all disjoint.
Define a new function $g: X \rightarrow Y$ such that $g(x) = f(x)$ on all $x \in E_n$ and on $x \in (\cup_{i=0}^\infty E_i)^c$ to be any
value. Then $g$ is $\mathcal{F}$-measurable and is a.e. equal to $f$.

I hope this proof is correct.

format: plain


text:
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Search Results: 1 - 10 of 100 matches for " "
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 Physics , 2013, DOI: 10.1088/0004-637X/787/2/138 Abstract: We present optical spectroscopy and optical/near-IR photometry of 31 host galaxies of hydrogen-poor superluminous supernovae (SLSNe), including 15 events from the Pan-STARRS1 Medium Deep Survey. Our sample spans the redshift range 0.1 < z < 1.6 and is the first comprehensive host galaxy study of this specific subclass of cosmic explosions. Combining the multi-band photometry and emission-line measurements, we determine the luminosities, stellar masses, star formation rates and metallicities. We find that as a whole, the hosts of SLSNe are a low-luminosity ( ~ -17.3 mag), low stellar mass ( ~ 2 x 10^8 M_sun) population, with a high median specific star formation rate ( ~ 2 Gyr^-1). The median metallicity of our spectroscopic sample is low, 12 + log(O/H}) ~ 8.35 ~ 0.45 Z_sun, although at least one host galaxy has solar metallicity. The host galaxies of H-poor SLSNe are statistically distinct from the hosts of GOODS core-collapse SNe (which cover a similar redshift range), but resemble the host galaxies of long-duration gamma-ray bursts (LGRBs) in terms of stellar mass, SFR, sSFR and metallicity. This result indicates that the environmental causes leading to massive stars forming either SLSNe or LGRBs are similar, and in particular that SLSNe are more effectively formed in low metallicity environments. We speculate that the key ingredient is large core angular momentum, leading to a rapidly-spinning magnetar in SLSNe and an accreting black hole in LGRBs.
 Physics , 2010, DOI: 10.1051/0004-6361/200913753 Abstract: Aims: We study the distribution of Wolf-Rayet (WR) stars and their subtypes with respect to their host galaxy light distribution. We thus want to investigate whether WR stars are potential progenitors of stripped-envelope core-collapse supernovae (SNe) and/or long-duration gamma-ray bursts (LGRBs). Methods: We derived the relative surface brightness (fractional flux) at the locations of WR stars and compared with similar results for LGRBs and SNe. We examined two nearby galaxies, M 83 and NGC 1313, for which a comprehensive study of the WR population exists. These two galaxies contain a sufficiently large number of WR stars and sample different metallicities. To enable the comparison, the images of the galaxies were processed to make them appear as they would look at a higher redshift. The robustness of our results against several sources of uncertainty was investigated with the aid of Monte Carlo simulations. Results: We find that the WC star distribution favours brighter pixels than the WN star population. WC stars are more likely drawn from the same distribution as SNe Ic than from other SN distributions, while WN stars show a higher degree of association with SNe Ib. It can also not be excluded that WR (especially WC) stars are related to LGRBs. Some differences between the two galaxies do exist, especially in the subtype distributions, and may stem from differences in metallicity. Conclusions: Although a conclusive answer is not possible, the expectation that WR stars are the progenitors of SNe Ib/c and LGRBs survives this test. The trend observed between the distributions of WN and WC stars, as compared to those of SNe Ib and Ic, is consistent with the theoretical picture that SNe Ic result from progenitors that have been stripped of a larger part of their envelope.
 Physics , 1998, DOI: 10.1086/311634 Abstract: The recent discovery of a supernova (SN 1998bw) seemingly associated with GRB~980425 adds a new twist to the decades-old debate over the origin of gamma-ray bursts. To investigate the possibility that some (or all) bursts are associated with supernovae, we performed a systematic search for temporal/angular correlations using catalogs of BATSE and BATSE/{\it Ulysses} burst locations. We find no associations with any of the precise BATSE/{\it Ulysses} locations, which allows us to conclude that the fraction of high-fluence gamma-ray bursts from known supernovae is small ($<$0.2%). For the more numerous weaker bursts, the corresponding limiting fraction of 1.5% is less constraining due to the imprecise locations of these events. This limit ($1.5% \simeq 18$ bursts) allows that a large fraction of the recent supernovae used as a comparison data set (18 supernovae $\simeq 20%$) could have associated gamma-ray bursts. Thus, although we find no significant evidence to support a burst/supernova association, the possibility cannot be excluded for weak bursts.
 J. Craig Wheeler Physics , 1999, Abstract: There are hints that nearby Type Ia supernovae may be a little different than those at large redshift. Confidence in the conclusion that there is a cosmological constant and an accelerating Universe thus still requires the hard work of sorting out potential systematic effects. Polarization data show that core-collapse supernovae (Type II and Ib/c) probably depart strongly from spherical symmetry. Evidence for exceedingly energetic supernovae must be considered self-consistently with evidence that they are asymmetric, a condition that affects energy estimates. Jets arising near the compact object can produce such asymmetries. There is growing conviction that gamma-ray bursts intrinsically involve collimated or jet-like flow and hence that they are also strongly asymmetric. SN 1998bw is a potential rosetta stone that will help to sort out the physics of explosive events. Are events like SN 1998bw more closely related to "ordinary" supernovae or "hypernovae?" Do they leave behind neutron stars as "ordinary" pulsars or "magnetars" or is the remnant a black hole? Are any of these events associated with classic cosmic gamma-ray bursts as suggested by the supernova-like modulation of the afterglows of GRB 970228, GRB 980326 and GRB 990712?
 Physics , 1999, Abstract: We use a rigorous method, based on Bayesian inference, for calculating the odds favoring the hypothesis that any particular class of astronomical transients produce gamma-ray bursts over the hypothesis that they do not. We then apply this method to a sample of 83 Type Ia supernovae and a sample of 20 Type Ib-Ic supernovae. We find overwhelming odds against the hypothesis that all Type Ia supernovae produce gamma-ray bursts, whether at low redshift ($10^{9}:1$) or high-redshift ($10^{12}:1$), and very large odds ($6000:1$) against the hypothesis that all Type Ib, Ib/c, and Ic supernovae produce observable gamma-ray bursts. We find large odds ($34:1$) against the hypothesis that a fraction of Type Ia supernovae produce observable gamma-ray bursts, and moderate odds ($6:1$) against the hypothesis that a fraction of Type Ib-Ic supernovae produce observable bursts.
 Physics , 2008, DOI: 10.1140/epjc/s10052-009-1086-z Abstract: Recently, Gamma-Ray Bursts (GRBs) were proposed to be a complementary cosmological probe to type Ia supernovae (SNIa). GRBs have been advocated to be standard candles since several empirical GRB luminosity relations were proposed as distance indicators. However, there is a so-called circularity problem in the direct use of GRBs. Recently, a new idea to calibrate GRBs in a completely cosmology independent manner has been proposed, and the circularity problem can be solved. In the present work, following the method proposed by Liang {\it et al.}, we calibrate 70 GRBs with the Amati relation using 307 SNIa. Then, following the method proposed by Shafieloo {\it et al.}, we smoothly reconstruct the cosmic expansion history up to redshift $z=6.29$ with the calibrated GRBs. We find some new features in the reconstructed results.
 Physics , 2009, DOI: 10.1016/j.nuclphysbps.2009.07.024 Abstract: A new method to constrain the cosmological equation of state is proposed by using combined samples of gamma-ray bursts (GRBs) and supernovae (SNeIa). The Chevallier-Polarski-Linder parameterization is adopted for the equation of state in order to find out a realistic approach to achieve the deceleration/acceleration transition phase of dark energy models. As results, we find that GRBs, calibrated by SNeIa, could be, at least, good distance indicators capable of discriminating cosmological models with respect to $\Lambda$CDM at high redshift.
 Physics , 1998, DOI: 10.1051/aas:1999313 Abstract: We present a rigorous method, based on Bayesian inference, for calculating the odds favoring the hypothesis that any particular class of astronomical transients produce gamma-ray bursts over the hypothesis that they do not. We then apply this method to a sample of 83 Type Ia supernovae and a sample of 20 Type Ib-Ic supernovae. We find overwhelming odds against the hypothesis that all Type Ia supernovae produce gamma-ray bursts, whether at low redshift ($10^{9}:1$) or high-redshift ($10^{12}:1$), and very large odds ($6000:1$) against the hypothesis that all Type Ib, Ib/c, and Ic supernovae produce observable gamma-ray bursts. We find large odds ($34:1$) against the hypothesis that a fraction of Type Ia supernovae produce observable gamma-ray bursts, and moderate odds ($6:1$) against the hypothesis that a fraction of Type Ib-Ic supernovae produce observable bursts. We have also re-analyzed both a corrected version of the Wang & Wheeler sample of Type Ib-Ic SNe and our larger sample of 20 Type Ib-Ic SNe, using a generalization of their frequentist method. We find no significant evidence in either case of a correlation between Type Ib-Ic SNe and GRBs, consistent with the very strong evidence against such a correlation that we find from our Bayesian analysis.
 Physics , 2013, DOI: 10.1088/0004-637X/773/1/53 Abstract: This is the first release of optical spectroscopic data of low-redshift Type Ia supernovae (SNe Ia) by the Carnegie Supernova Project including 604 previously unpublished spectra of 93 SNe Ia. The observations cover a range of phases from 12 days before to over 150 days after the time of B-band maximum light. With the addition of 228 near-maximum spectra from the literature we study the diversity among SNe Ia in a quantitative manner. For that purpose, spectroscopic parameters are employed such as expansion velocities from spectral line blueshifts, and pseudo-equivalent widths (pW). The values of those parameters at maximum light are obtained for 78 objects, thus providing a characterization of SNe Ia that may help to improve our understanding of the properties of the exploding systems and the thermonuclear flame propagation. Two objects, namely SNe 2005M and 2006is, stand out from the sample by showing peculiar Si II and S II velocities but otherwise standard velocities for the rest of the ions. We further study the correlations between spectroscopic and photometric parameters such as light-curve decline rate and color. In agreement with previous studies, we find that the pW of Si II absorption features are very good indicators of light-curve decline rate. Furthermore, we demonstrate that parameters such as pW2(SiII4130) and pW6(SiII5972) provide precise calibrations of the peak B-band luminosity with dispersions of ~0.15 mag. In the search for a secondary parameter in the calibration of peak luminosity for SNe Ia, we find a ~2--3-sigma correlation between B-band Hubble residuals and the velocity at maximum light of S II and Si II lines.
 Physics , 1998, DOI: 10.1086/305723 Abstract: Long-term monitoring of the radio emission from supernovae with the Very Large Array (VLA) shows that the radio light curves'' evolve in a systematic fashion with a distinct peak flux density (and thus, in combination with a distance, a peak spectral luminosity) at each frequency and a well-defined time from explosion to that peak. Studying these two quantities at 6 cm wavelength, peak spectral luminosity (L_{6 cm peak}) and time after explosion date (t_0) to reach that peak (t_{6 cm peak} - t_0), we find that they appear related. In particular, based on two objects, Type Ib supernovae may be approximate radio standard candles'' with a 6 cm peak luminosity L_{6 cm peak} \approx 19.9 X 10^{26} erg s^{-1} Hz^{-1}; also based on two objects, Type Ic supernovae may be approximate radio standard candles'' with a 6 cm peak luminosity L_{6 cm peak} \approx 6.5 X 10^{26} erg s^{-1} Hz^{-1}; and, based on twelve objects, Type II supernovae appear to obey a relation L_{6 cm peak} \simeq 5.5 X 10^{23} (t_{6 cm peak} - t_0)^{1.4} erg s^{-1} Hz^{-1}, with time measured in days. If these relations are supported by further observations, they provide a means for determining distances to supernovae, and thus to their parent galaxies, from purely radio continuum observations.
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A Man Walks into a Bar

A cache by dgauss Message this owner
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Geocache Description:

Spiromania #40

This is a puzzle cache so it is not located at the posed coordinates. Rather it is at VKabcdefghij which is short for UTM: 15T E 4abcde N 49fghij.

• A geocacher.
• A geocacher walks into a bar and his dog walked under it.
• A man walks into the bar and sits by two geocachers, a barfly and a pickpocket.
• A man stomps into the bar, slurps a whiskey, and guzzles a beer.
• A pirate with a patch over his eye named Clyde walks into the bar.
• After one drink, the man took his change and his leave.
• Having had a big lunch, the bar didn't seem all that appealing.
• Slow as molasses on a cucumber, a man slithers into the bar.
• Suffering from very close veins, a man ambers into the bar and orders a rooster tail.
• A man walks into the bar, sits alone in the crowd, and has a dry martini.
• A man swims up to the Tiki bar beside the pool on his cruise ship in the Caribbean. It's blue and has an olive and a lot of alcohol in it.
• A man walks into the bar with a parrot. Some T5s are fake. The butler did it in the conservatory with the candlestick.

V) a portmanteau
K) mixed tenses
0) onomatopoeis
1) oxymorons
2) mixed metaphor
3) misplaced modifier
4) non sequitur
5) malapropisms
6) zeugma
7) Oxford comma missing
8) antecedent unclear
9) dangling modifier

Enter your answer into the coordinate checker in the form VKabcdefghij, where abcdefghij is a permutation of 0123456789, which makes this cache qualify for the dg-point challenge.

You can validate your puzzle solution with certitude.

Congrats to the first 10 solvers:

1.      Ms Sunshine     Sat, 24 Mar 2018 9:06:50
2.      ctc128  Sat, 24 Mar 2018 15:17:23
3.      pfalstad        Sat, 24 Mar 2018 16:41:13
4.      teamajk Mon, 26 Mar 2018 22:49:36
5.      sparkyfry       Tue, 27 Mar 2018 13:24:19
6.      the Dino-mic Duo        Tue, 27 Mar 2018 16:29:45
7.      mnboyscout      Wed, 28 Mar 2018 15:29:43
8.      salsman Wed, 28 Mar 2018 18:40:01